Formula For Pdf Of W Y \mathbf{W}\mathbf{y} Wy Where Y \mathbf{y} Y Is On Unit Sphere

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Introduction

In probability theory and statistics, the problem of finding the probability density function (PDF) of a linearly transformed random variable is a common and important task. Given a random variable y\mathbf{y} defined on the unit sphere embedded in R3\mathbb{R}^3, and the linearly transformed variable z=Wy\mathbf{z} = \mathbf{W}\mathbf{y}, where W\mathbf{W} is an orthogonal matrix, we aim to derive the formula for the PDF of z\mathbf{z}.

Background

To tackle this problem, we need to recall some fundamental concepts in probability theory and linear algebra. The unit sphere in R3\mathbb{R}^3 is defined as the set of points y\mathbf{y} such that ∥y∥2=1\|\mathbf{y}\|_2 = 1, where ∥⋅∥2\|\cdot\|_2 denotes the Euclidean norm. An orthogonal matrix W\mathbf{W} satisfies the property WTW=I\mathbf{W}^T\mathbf{W} = \mathbf{I}, where I\mathbf{I} is the identity matrix.

Derivation of the PDF

Let y\mathbf{y} be a random variable defined on the unit sphere, and let z=Wy\mathbf{z} = \mathbf{W}\mathbf{y} be the linearly transformed variable. We want to find the PDF of z\mathbf{z}, denoted as fz(x)f_{\mathbf{z}}(\mathbf{x}). To do this, we can use the change of variables formula, which states that the PDF of z\mathbf{z} can be expressed as:

fz(x)=fy(W−1x)⋅∣det⁡(W)∣f_{\mathbf{z}}(\mathbf{x}) = f_{\mathbf{y}}(\mathbf{W}^{-1}\mathbf{x}) \cdot |\det(\mathbf{W})|

where fy(y)f_{\mathbf{y}}(\mathbf{y}) is the PDF of y\mathbf{y}, and det⁡(W)\det(\mathbf{W}) is the determinant of the matrix W\mathbf{W}.

Since W\mathbf{W} is an orthogonal matrix, we have det⁡(W)=±1\det(\mathbf{W}) = \pm 1. Moreover, since y\mathbf{y} is defined on the unit sphere, we can assume that fy(y)=g(∥y∥2)f_{\mathbf{y}}(\mathbf{y}) = g(\|\mathbf{y}\|_2), where gg is a function that depends only on the norm of y\mathbf{y}.

Using the change of variables formula, we can rewrite the PDF of z\mathbf{z} as:

fz(x)=g(∥W−1x∥2)⋅∣det⁡(W)∣f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{W}^{-1}\mathbf{x}\|_2) \cdot |\det(\mathbf{W})|

Since W\mathbf{W} is orthogonal, we have ∥W−1x∥2=∥x∥2\|\mathbf{W}^{-1}\mathbf{x}\|_2 = \|\mathbf{x}\|_2. Therefore, the PDF of z\mathbf{z} can be simplified as:

fz(x)=g(∥x∥2)⋅∣det⁡(W)∣f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot |\det(\mathbf{W})|

Jacobian Determinant

To find the PDF of z\mathbf{z}, we need to evaluate the Jacobian determinant of the transformation z=Wy\mathbf{z} = \mathbf{W}\mathbf{y}. The Jacobian matrix of this transformation is given by:

J=∂z∂y=W\mathbf{J} = \frac{\partial \mathbf{z}}{\partial \mathbf{y}} = \mathbf{W}

The Jacobian determinant is then given by:

det⁡(J)=det⁡(W)\det(\mathbf{J}) = \det(\mathbf{W})

Since W\mathbf{W} is orthogonal, we have det⁡(W)=±1\det(\mathbf{W}) = \pm 1. Therefore, the Jacobian determinant is:

det⁡(J)=±1\det(\mathbf{J}) = \pm 1

PDF of z\mathbf{z}

Using the change of variables formula, we can express the PDF of z\mathbf{z} as:

fz(x)=fy(W−1x)⋅∣det⁡(W)∣f_{\mathbf{z}}(\mathbf{x}) = f_{\mathbf{y}}(\mathbf{W}^{-1}\mathbf{x}) \cdot |\det(\mathbf{W})|

Since W\mathbf{W} is orthogonal, we have det⁡(W)=±1\det(\mathbf{W}) = \pm 1. Moreover, since y\mathbf{y} is defined on the unit sphere, we can assume that fy(y)=g(∥y∥2)f_{\mathbf{y}}(\mathbf{y}) = g(\|\mathbf{y}\|_2), where gg is a function that depends only on the norm of y\mathbf{y}.

Using the Jacobian determinant, we can rewrite the PDF of z\mathbf{z} as:

fz(x)=g(∥x∥2)⋅∣det⁡(W)∣f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot |\det(\mathbf{W})|

Since det⁡(W)=±1\det(\mathbf{W}) = \pm 1, we have:

fz(x)=g(∥x∥2)⋅±1f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot \pm 1

Conclusion

In this article, we have derived the formula for the PDF of z=Wy\mathbf{z} = \mathbf{W}\mathbf{y}, where y\mathbf{y} is a random variable defined on the unit sphere, and W\mathbf{W} is an orthogonal matrix. We have shown that the PDF of z\mathbf{z} can be expressed as:

fz(x)=g(∥x∥2)⋅±1f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot \pm 1

where gg is a function that depends only on the norm of y\mathbf{y}, and ±1\pm 1 is the Jacobian determinant of the transformation z=Wy\mathbf{z} = \mathbf{W}\mathbf{y}.

References

  • [1] Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
  • [2] Kendall, D. G. (1957). The Solution of a System of Linear Equations by Minimization. Proceedings of the Cambridge Philosophical Society, 53, 67-72.
  • [3] Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press.

Appendix

A.1 Jacobian Matrix

The Jacobian matrix of the transformation z=Wy\mathbf{z} = \mathbf{W}\mathbf{y} is given by:

J=∂z∂y=W\mathbf{J} = \frac{\partial \mathbf{z}}{\partial \mathbf{y}} = \mathbf{W}

A.2 Jacobian Determinant

The Jacobian determinant is given by:

det⁡(J)=det⁡(W)\det(\mathbf{J}) = \det(\mathbf{W})

Since W\mathbf{W} is orthogonal, we have det⁡(W)=±1\det(\mathbf{W}) = \pm 1. Therefore, the Jacobian determinant is:

\det(\mathbf{J}) = \pm 1$<br/> **Q&A: Formula for the PDF of $\mathbf{W}\mathbf{y}$ where $\mathbf{y}$ is on the Unit Sphere** ===========================================================

Q: What is the formula for the PDF of z=Wy\mathbf{z} = \mathbf{W}\mathbf{y}, where y\mathbf{y} is a random variable defined on the unit sphere, and W\mathbf{W} is an orthogonal matrix?

A: The formula for the PDF of z\mathbf{z} is given by:

fz(x)=g(∥x∥2)⋅±1</span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotationencoding="application/x−tex">g</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span></span></span></span>isafunctionthatdependsonlyonthenormof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>,and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>istheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>.</p><h2><strong>Q:WhatistheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>?</strong></h2><p>A:TheJacobiandeterminantisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">det⁡(J)=det⁡(W)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span></span></span></span></span></p><p>Since<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isorthogonal,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(W)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>.Therefore,theJacobiandeterminantis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(J)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span></span></p><h2><strong>Q:HowdoIfindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>when<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>isarandomvariabledefinedontheunitsphere?</strong></h2><p>A:TofindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>,youneedtoevaluatetheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>.TheJacobiandeterminantisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">det⁡(J)=det⁡(W)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span></span></span></span></span></p><p>Since<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isorthogonal,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(W)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>.Therefore,theJacobiandeterminantis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(J)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span></span></p><h2><strong>Q:WhatistherelationshipbetweenthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>andthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>?</strong></h2><p>A:ThePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>canbeexpressedas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>f</mi><mimathvariant="bold">z</mi></msub><mostretchy="false">(</mo><mimathvariant="bold">x</mi><mostretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mimathvariant="bold">y</mi></msub><mostretchy="false">(</mo><msup><mimathvariant="bold">W</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mimathvariant="bold">x</mi><mostretchy="false">)</mo><mo>⋅</mo><mimathvariant="normal">∣</mi><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mimathvariant="normal">∣</mi></mrow><annotationencoding="application/x−tex">fz(x)=fy(W−1x)⋅∣det⁡(W)∣</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1611em;"><spanstyle="top:−2.55em;margin−left:−0.1076em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathbfmtight">z</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathbf">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1502em;vertical−align:−0.2861em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1611em;"><spanstyle="top:−2.55em;margin−left:−0.1076em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathbfmtight"style="margin−right:0.01597em;">y</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2861em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mord"><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">−</span><spanclass="mordmtight">1</span></span></span></span></span></span></span></span></span><spanclass="mordmathbf">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">∣</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mord">∣</span></span></span></span></span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mimathvariant="bold">y</mi></msub><mostretchy="false">(</mo><mimathvariant="bold">y</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">fy(y)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1.0361em;vertical−align:−0.2861em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1611em;"><spanstyle="top:−2.55em;margin−left:−0.1076em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathbfmtight"style="margin−right:0.01597em;">y</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.2861em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span><spanclass="mclose">)</span></span></span></span>isthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>,and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">det⁡(W)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span></span></span></span>isthedeterminantofthematrix<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>.</p><h2><strong>Q:HowdoIusethechangeofvariablesformulatofindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>?</strong></h2><p>A:Tousethechangeofvariablesformula,youneedtoevaluatetheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>.TheJacobiandeterminantisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">det⁡(J)=det⁡(W)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span></span></span></span></span></p><p>Since<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isorthogonal,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(W)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>.Therefore,theJacobiandeterminantis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(J)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span></span></p><h2><strong>Q:WhatisthesignificanceoftheJacobiandeterminantinthechangeofvariablesformula?</strong></h2><p>A:TheJacobiandeterminantisusedtoscalethePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>toobtainthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>.TheJacobiandeterminantisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">det⁡(J)=det⁡(W)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span></span></span></span></span></p><p>Since<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isorthogonal,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(W)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>.Therefore,theJacobiandeterminantis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(J)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span></span></p><h2><strong>Q:CanIusethechangeofvariablesformulatofindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>when<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>isarandomvariabledefinedonasphereofradius<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotationencoding="application/x−tex">r</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span></span></span></span>?</strong></h2><p>A:Yes,youcanusethechangeofvariablesformulatofindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>when<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>isarandomvariabledefinedonasphereofradius<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotationencoding="application/x−tex">r</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span></span></span></span>.TheonlydifferenceisthattheJacobiandeterminantwillbe:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotationencoding="application/x−tex">det⁡(J)=r2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p><h2><strong>Q:HowdoIfindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>when<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>isarandomvariabledefinedonasphereofradius<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotationencoding="application/x−tex">r</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span></span></span></span>and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isanorthogonalmatrix?</strong></h2><p>A:TofindthePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>,youneedtoevaluatetheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>.TheJacobiandeterminantisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>⋅</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotationencoding="application/x−tex">det⁡(J)=det⁡(W)⋅r2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8641em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p>Since<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">W</mi></mrow><annotationencoding="application/x−tex">W</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6861em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span></span></span></span>isorthogonal,wehave<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">W</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">det⁡(W)=±1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf"style="margin−right:0.01597em;">W</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord">1</span></span></span></span>.Therefore,theJacobiandeterminantis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mostretchy="false">(</mo><mimathvariant="bold">J</mi><mostretchy="false">)</mo><mo>=</mo><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotationencoding="application/x−tex">det⁡(J)=±r2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">det</span><spanclass="mopen">(</span><spanclass="mordmathbf">J</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.9474em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p>ThePDFof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi></mrow><annotationencoding="application/x−tex">z</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span></span></span></span>canbeexpressedas:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><msub><mi>f</mi><mimathvariant="bold">z</mi></msub><mostretchy="false">(</mo><mimathvariant="bold">x</mi><mostretchy="false">)</mo><mo>=</mo><mi>g</mi><mostretchy="false">(</mo><mimathvariant="normal">∥</mi><mimathvariant="bold">x</mi><msub><mimathvariant="normal">∥</mi><mn>2</mn></msub><mostretchy="false">)</mo><mo>⋅</mo><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotationencoding="application/x−tex">fz(x)=g(∥x∥2)⋅±r2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.1611em;"><spanstyle="top:−2.55em;margin−left:−0.1076em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmathbfmtight">z</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mopen">(</span><spanclass="mordmathbf">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span><spanclass="mopen">(</span><spanclass="mord">∥</span><spanclass="mordmathbf">x</span><spanclass="mord"><spanclass="mord">∥</span><spanclass="msupsub"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.3011em;"><spanstyle="top:−2.55em;margin−left:0em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.15em;"><span></span></span></span></span></span></span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">⋅</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.9474em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotationencoding="application/x−tex">g</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">g</span></span></span></span>isafunctionthatdependsonlyonthenormof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">y</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6389em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">y</span></span></span></span>,and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotationencoding="application/x−tex">±r2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8974em;vertical−align:−0.0833em;"></span><spanclass="mord">±</span><spanclass="mord"><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8141em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span></span></span></span>istheJacobiandeterminantofthetransformation<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mimathvariant="bold">z</mi><mo>=</mo><mimathvariant="bold">W</mi><mimathvariant="bold">y</mi></mrow><annotationencoding="application/x−tex">z=Wy</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4444em;"></span><spanclass="mordmathbf">z</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8805em;vertical−align:−0.1944em;"></span><spanclass="mordmathbf"style="margin−right:0.01597em;">Wy</span></span></span></span>.</p>f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot \pm 1 </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span> is a function that depends only on the norm of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span> is the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>.</p> <h2><strong>Q: What is the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>?</strong></h2> <p>A: The Jacobian determinant is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \det(\mathbf{W}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span></span></span></span></span></p> <p>Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is orthogonal, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{W}) = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span>. Therefore, the Jacobian determinant is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \pm 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p> <h2><strong>Q: How do I find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> is a random variable defined on the unit sphere?</strong></h2> <p>A: To find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span>, you need to evaluate the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>. The Jacobian determinant is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \det(\mathbf{W}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span></span></span></span></span></p> <p>Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is orthogonal, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{W}) = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span>. Therefore, the Jacobian determinant is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \pm 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p> <h2><strong>Q: What is the relationship between the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> and the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span>?</strong></h2> <p>A: The PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> can be expressed as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi mathvariant="bold">z</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mi mathvariant="bold">y</mi></msub><mo stretchy="false">(</mo><msup><mi mathvariant="bold">W</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f_{\mathbf{z}}(\mathbf{x}) = f_{\mathbf{y}}(\mathbf{W}^{-1}\mathbf{x}) \cdot |\det(\mathbf{W})| </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1502em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight" style="margin-right:0.01597em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathbf">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi mathvariant="bold">y</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_{\mathbf{y}}(\mathbf{y})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight" style="margin-right:0.01597em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">y</span><span class="mclose">)</span></span></span></span> is the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(\mathbf{W})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span></span></span></span> is the determinant of the matrix <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span>.</p> <h2><strong>Q: How do I use the change of variables formula to find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span>?</strong></h2> <p>A: To use the change of variables formula, you need to evaluate the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>. The Jacobian determinant is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \det(\mathbf{W}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span></span></span></span></span></p> <p>Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is orthogonal, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{W}) = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span>. Therefore, the Jacobian determinant is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \pm 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p> <h2><strong>Q: What is the significance of the Jacobian determinant in the change of variables formula?</strong></h2> <p>A: The Jacobian determinant is used to scale the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> to obtain the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span>. The Jacobian determinant is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \det(\mathbf{W}) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span></span></span></span></span></p> <p>Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is orthogonal, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{W}) = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span>. Therefore, the Jacobian determinant is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \pm 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p> <h2><strong>Q: Can I use the change of variables formula to find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> is a random variable defined on a sphere of radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>?</strong></h2> <p>A: Yes, you can use the change of variables formula to find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> is a random variable defined on a sphere of radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>. The only difference is that the Jacobian determinant will be:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = r^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p> <h2><strong>Q: How do I find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> when <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span> is a random variable defined on a sphere of radius <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is an orthogonal matrix?</strong></h2> <p>A: To find the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span>, you need to evaluate the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>. The Jacobian determinant is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \det(\mathbf{W}) \cdot r^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p> <p>Since <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6861em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">W</span></span></span></span> is orthogonal, we have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">W</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\det(\mathbf{W}) = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span>. Therefore, the Jacobian determinant is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi mathvariant="bold">J</mi><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\det(\mathbf{J}) = \pm r^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathbf">J</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p> <p>The PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi></mrow><annotation encoding="application/x-tex">\mathbf{z}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span></span></span></span> can be expressed as:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi mathvariant="bold">z</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∥</mi><mi mathvariant="bold">x</mi><msub><mi mathvariant="normal">∥</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f_{\mathbf{z}}(\mathbf{x}) = g(\|\mathbf{x}\|_2) \cdot \pm r^2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1611em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathbf mtight">z</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathbf">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">∥</span><span class="mord mathbf">x</span><span class="mord"><span class="mord">∥</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span> is a function that depends only on the norm of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">y</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>±</mo><msup><mi>r</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\pm r^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> is the Jacobian determinant of the transformation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">z</mi><mo>=</mo><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{z} = \mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4444em;"></span><span class="mord mathbf">z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8805em;vertical-align:-0.1944em;"></span><span class="mord mathbf" style="margin-right:0.01597em;">Wy</span></span></span></span>.</p>