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, linear transformations play a crucial role in understanding the behavior of random variables. When a random variable $\mathbf{y}$ is linearly transformed by a matrix $\mathbf{W}$, the resulting variable $\mathbf{z} = \mathbf{W}\mathbf{y}$ inherits the properties of both the original variable and the transformation matrix. In this article, we will focus on deriving the probability density function (PDF) of the linearly transformed variable $\mathbf{z}$, where $\mathbf{y}$ is defined on the unit sphere embedded in $\mathbb{R}^3$.

Probability Distributions and Linear Transformations

Probability distributions are used to describe the behavior of random variables, and linear transformations are a fundamental concept in understanding how these variables change under different operations. When a random variable $\mathbf{y}$ is linearly transformed by a matrix $\mathbf{W}$, the resulting variable $\mathbf{z} = \mathbf{W}\mathbf{y}$ has a probability distribution that is related to the original distribution of $\mathbf{y}$.

Jacobian and Change of Variables

The Jacobian matrix plays a crucial role in understanding how the probability distribution of a random variable changes under a linear transformation. The Jacobian matrix $\mathbf{J}$ of a linear transformation $\mathbf{z} = \mathbf{W}\mathbf{y}$ is given by:

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

The Jacobian matrix $\mathbf{J}$ is used to compute the change of variables, which is essential in deriving the PDF of the linearly transformed variable $\mathbf{z}$.

Deriving the PDF of $\mathbf{W}\mathbf{y}$

To derive the PDF of $\mathbf{z} = \mathbf{W}\mathbf{y}$, we need to consider the joint PDF of $\mathbf{y}$ and the determinant of the Jacobian matrix $\mathbf{J}$. The joint PDF of $\mathbf{y}$ is given by:

$$f_{\mathbf{y}}(\mathbf{y}) = \frac{1}{A}$$

where $A$ is the surface area of the unit sphere.

The determinant of the Jacobian matrix $\mathbf{J}$ is given by:

$$|\mathbf{J}| = |\mathbf{W}|$$

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

$$f_{\mathbf{z}}(\mathbf{z}) = f_{\mathbf{y}}(\mathbf{y}) \cdot |\mathbf{J}|$$

Substituting the expressions for $f_{\mathbf{y}}(\mathbf{y})$ and $|\mathbf{J}|$, we get:

$$f_{\mathbf{z}}(\mathbf{z}) = \frac{1}{A} \cdot |\mathbf{W}|$$

Properties of the PDF of $\mathbf{W}\mathbf{y}$

The PDF of $\mathbf{z} = \mathbf{W}\mathbf{y}$ has several important properties that are worth noting:

• Non-negativity: The PDF of $\mathbf{z}$ is non-negative for all values of $\mathbf{z}$.
• Normalization: The PDF of $\mathbf{z}$ is normalized, meaning that the integral of the PDF over the entire support of $\mathbf{z}$ is equal to 1.
• Invariance: The PDF of $\mathbf{z}$ is invariant under linear transformations, meaning that the PDF of $\mathbf{z}$ remains the same under a change of variables.

Conclusion

In this article, we have derived the formula for the PDF of $\mathbf{W}\mathbf{y}$ where $\mathbf{y}$ is on the unit sphere. We have also discussed the properties of the PDF and its relation to the Jacobian matrix and the change of variables. The formula for the PDF of $\mathbf{W}\mathbf{y}$ is given by:

$$f_{\mathbf{z}}(\mathbf{z}) = \frac{1}{A} \cdot |\mathbf{W}|$$

where $A$ is the surface area of the unit sphere and $|\mathbf{W}|$ is the determinant of the Jacobian matrix.

References

• [1] Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
• [2] Kendall, D. G. (1957). The Transformation of Three-Space Forms. Biometrika, 44(3/4), 257-268.
• [3] Mardia, K. V. (1972). Statistics of Directional Data. Academic Press.

Appendix

A. Derivation of the Jacobian Matrix

The Jacobian matrix $\mathbf{J}$ of a linear transformation $\mathbf{z} = \mathbf{W}\mathbf{y}$ is given by:

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

B. Derivation of the PDF of $\mathbf{W}\mathbf{y}$

To derive the PDF of $\mathbf{z} = \mathbf{W}\mathbf{y}$, we need to consider the joint PDF of $\mathbf{y}$ and the determinant of the Jacobian matrix $\mathbf{J}$. The joint PDF of $\mathbf{y}$ is given by:

$$f_{\mathbf{y}}(\mathbf{y}) = \frac{1}{A}$$

where $A$ is the surface area of the unit sphere.

The determinant of the Jacobian matrix $\mathbf{J}$ is given by:

$$|\mathbf{J}| = |\mathbf{W}|$$

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

$$f_{\mathbf{z}}(\mathbf{z}) = f_{\mathbf{y}}(\mathbf{y}) \cdot |\mathbf{J}|$$

Substituting the expressions for $f_{\mathbf{y}}(\mathbf{y})$ and $|\mathbf{J}|$, we get:

f_{\mathbf{z}}(\mathbf{z}) = \frac{1}{A} \cdot |\mathbf{W}|$<br/> **Q&A: Formula for the PDF of $\mathbf{W}\mathbf{y}$ where $\mathbf{y}$ is on the Unit Sphere** =====================================================================================

Introduction

In our previous article, we derived the formula for the PDF of $\mathbf{W}\mathbf{y}$ where $\mathbf{y}$ is on the unit sphere. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the unit sphere?

A: The unit sphere is a three-dimensional sphere with a radius of 1. It is a closed surface that is centered at the origin and has a constant distance of 1 from the origin.

Q: What is the Jacobian matrix?

A: The Jacobian matrix is a matrix that represents the linear transformation of a function. In the context of the PDF of $\mathbf{W}\mathbf{y}$, the Jacobian matrix is given by $\mathbf{J} = \mathbf{W}$.

Q: What is the determinant of the Jacobian matrix?

A: The determinant of the Jacobian matrix is given by $|\mathbf{J}| = |\mathbf{W}|$. This determinant represents the scaling factor of the linear transformation.

Q: How do I compute the PDF of $\mathbf{W}\mathbf{y}$?

A: To compute the PDF of $\mathbf{W}\mathbf{y}$, you need to follow these steps:

1. Compute the joint PDF of $\mathbf{y}$, which is given by $f_{\mathbf{y}}(\mathbf{y}) = \frac{1}{A}$, where $A$ is the surface area of the unit sphere.
2. Compute the determinant of the Jacobian matrix, which is given by $|\mathbf{J}| = |\mathbf{W}|$.
3. Use the change of variables formula to derive the PDF of $\mathbf{z}$, which is given by $f_{\mathbf{z}}(\mathbf{z}) = f_{\mathbf{y}}(\mathbf{y}) \cdot |\mathbf{J}|$.

Q: What are the properties of the PDF of $\mathbf{W}\mathbf{y}$?

A: The PDF of $\mathbf{z} = \mathbf{W}\mathbf{y}$ has several important properties, including:

• Non-negativity: The PDF of $\mathbf{z}$ is non-negative for all values of $\mathbf{z}$.
• Normalization: The PDF of $\mathbf{z}$ is normalized, meaning that the integral of the PDF over the entire support of $\mathbf{z}$ is equal to 1.
• Invariance: The PDF of $\mathbf{z}$ is invariant under linear transformations, meaning that the PDF of $\mathbf{z}$ remains the same under a change of variables.

Q: Can I use this formula for other types of linear transformations?

A: Yes, you can use this formula for other types of linear transformations. However, you need to modify the formula to accommodate the specific type of linear transformation.

Q: What are some common applications of this formula?

A: This formula has several common applications in statistics and machine learning, including:

• Data transformation: This formula can be used to transform data from one distribution to another.
• Feature extraction: This formula can be used to extract features from data that are more informative than the original data.
• Dimensionality reduction: This formula can be used to reduce the dimensionality of data while preserving the essential information.

Conclusion

In this article, we have answered some frequently asked questions related to the formula for the PDF of $\mathbf{W}\mathbf{y}$ where $\mathbf{y}$ is on the unit sphere. We hope that this article has provided you with a better understanding of this topic and its applications.

References

• [1] Papoulis, A. (1984). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.
• [2] Kendall, D. G. (1957). The Transformation of Three-Space Forms. Biometrika, 44(3/4), 257-268.
• [3] Mardia, K. V. (1972). Statistics of Directional Data. Academic Press.

Appendix

A. Derivation of the Jacobian Matrix

The Jacobian matrix $\mathbf{J}$ of a linear transformation $\mathbf{z} = \mathbf{W}\mathbf{y}$ is given by:

$$\mathbf{J} = \frac{\partial \mathbf{z}}{\partial \mathbf{y}} = \mathbf{W} </span></p> <h3>B. Derivation of the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><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></h3> <p>To derive 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><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>, you need to follow these steps:</p> <ol> <li>Compute the joint 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>, which is given by <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><mo>=</mo><mfrac><mn>1</mn><mi>A</mi></mfrac></mrow><annotation encoding="application/x-tex">f_{\mathbf{y}}(\mathbf{y}) = \frac{1}{A}</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 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:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> is the surface area of the unit sphere.</li> <li>Compute the determinant of the Jacobian matrix, which is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="bold">J</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold">W</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\mathbf{J}| = |\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><span class="mord mathbf">J</span><span class="mord">∣</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">∣</span><span class="mord mathbf" style="margin-right:0.01597em;">W</span><span class="mord">∣</span></span></span></span>.</li> <li>Use the change of variables formula to derive 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>, which is given by <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">z</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi><mi mathvariant="bold">y</mi></msub><mo stretchy="false">(</mo><mi mathvariant="bold">y</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><mi mathvariant="bold">J</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f_{\mathbf{z}}(\mathbf{z}) = f_{\mathbf{y}}(\mathbf{y}) \cdot |\mathbf{J}|</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">z</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:1.0361em;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 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="mord mathbf">J</span><span class="mord">∣</span></span></span></span>.</li> </ol> <h3>C. Properties of the PDF of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">W</mi><mi mathvariant="bold">y</mi></mrow><annotation encoding="application/x-tex">\mathbf{W}\mathbf{y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><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></h3> <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><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> has several important properties, including:</p> <ul> <li>Non-negativity: 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> is non-negative for all values 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>.</li> <li>Normalization: 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> is normalized, meaning that the integral of the PDF over the entire support 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> is equal to 1.</li> <li>Invariance: 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> is invariant under linear transformations, meaning that 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> remains the same under a change of variables.</li> </ul>$$