How To Calculate E [ X 2 Y 2 ∣ X 2 + Y 2 = D ] E[X^2Y^2| X^2 +Y^2 = D] E [ X 2 Y 2 ∣ X 2 + Y 2 = D ] When X X X And Y Y Y Are Two Gaussian Random Variables

Introduction

In probability theory, the expectation of a function of random variables is a fundamental concept that has numerous applications in various fields, including statistics, engineering, and finance. When dealing with Gaussian random variables, the calculation of expectations can be simplified using the properties of the normal distribution. In this article, we will discuss how to calculate the conditional expectation of a quadratic form of two independent Gaussian random variables, given a condition on their sum of squares.

Background

Let $X$ and $Y$ be two independent Gaussian random variables with mean $x_0$ and $y_0$, respectively, and equal variance $\sigma^2$. We assume that $x_0^2 + y_0^2 < D$, where $D$ is a given constant. Our goal is to calculate the conditional expectation of $X^2Y^2$ given that $X^2 + Y^2 = D$.

Properties of Gaussian Random Variables

Before diving into the calculation, let's recall some important properties of Gaussian random variables:

• A Gaussian random variable $X$ with mean $\mu$ and variance $\sigma^2$ has a probability density function (PDF) given by: $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma2}\right)$
• The expectation of a function $g(X)$ of a Gaussian random variable $X$ is given by: $E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) dx$
• The covariance between two Gaussian random variables $X$ and $Y$ is given by: $Cov(X, Y) = E[(X-E[X])(Y-E[Y])]$

Conditional Expectation

To calculate the conditional expectation of $X^2Y^2$ given that $X^2 + Y^2 = D$, we can use the definition of conditional expectation:

$$E[X^2Y^2|X^2 + Y^2 = D] = \frac{E[X^2Y^2 \cdot I_{\{X^2 + Y^2 = D\}}]}{P(X^2 + Y^2 = D)}$$

where $I_{\{X^2 + Y^2 = D\}}$ is the indicator function that takes value 1 if $X^2 + Y^2 = D$ and 0 otherwise.

Simplifying the Calculation

Using the properties of Gaussian random variables, we can simplify the calculation of the conditional expectation:

• Since $X$ and $Y$ are independent, we can write: $E[X^2Y2|X^2 + Y^2 = D] = E[X^2|X2 + Y^2 = D] \cdot E[Y^2|X2 + Y^2 = D]$
• Using the definition of conditional expectation, we can write: $E[X^2|X2 + Y^2 = D] = \frac{E[X^2 \cdot I_{{X^2 + Y^2 = D}}]}{P(X^2 + Y^2 = D)}$

Calculating the Conditional Expectation

To calculate the conditional expectation, we need to find the joint PDF of $X$ and $Y$ given that $X^2 + Y^2 = D$. Using the properties of Gaussian random variables, we can write:

• The joint PDF of $X$ and $Y$ is given by: $f(x, y) = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{x^2 + y^2}{2\sigma2}\right)$
• The joint PDF of $X$ and $Y$ given that $X^2 + Y^2 = D$ is given by: $f(x, y|X^2 + Y^2 = D) = \frac{f(x, y)}{P(X^2 + Y^2 = D)}$

Using the Bivariate Normal Distribution

Since $X$ and $Y$ are Gaussian random variables, their joint distribution is a bivariate normal distribution. Using the properties of the bivariate normal distribution, we can write:

• The joint PDF of $X$ and $Y$ is given by: $f(x, y) = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{x^2 + y^2 - 2\rho xy}{2\sigma^2(1-\rho2)}\right)$
• The joint PDF of $X$ and $Y$ given that $X^2 + Y^2 = D$ is given by: $f(x, y|X^2 + Y^2 = D) = \frac{f(x, y)}{P(X^2 + Y^2 = D)}$

Calculating the Conditional Expectation using the Bivariate Normal Distribution

Using the properties of the bivariate normal distribution, we can calculate the conditional expectation as follows:

• The conditional expectation of $X^2$ given that $X^2 + Y^2 = D$ is given by: $E[X^2|X2 + Y^2 = D] = \frac{\sigma^2(1-\rho2) + \rho^2 x_0^2}{\sigma2(1-\rho^2) + x_0^2} x_0^2$
• The conditional expectation of $Y^2$ given that $X^2 + Y^2 = D$ is given by: $E[Y^2|X2 + Y^2 = D] = \frac{\sigma^2(1-\rho2) + \rho^2 y_0^2}{\sigma2(1-\rho^2) + y_0^2} y_0^2$

Final Calculation

Using the results from the previous section, we can calculate the final result as follows:

• The conditional expectation of $X^2Y^2$ given that $X^2 + Y^2 = D$ is given by: $E[X^2Y2|X^2 + Y^2 = D] = E[X^2|X2 + Y^2 = D] \cdot E[Y^2|X2 + Y^2 = D]$

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to the conditional expectation of a quadratic form of two independent Gaussian random variables, given a condition on their sum of squares.

Q: What is the conditional expectation of $X^2Y^2$ given that $X^2 + Y^2 = D$?

A: The conditional expectation of $X^2Y^2$ given that $X^2 + Y^2 = D$ is given by the product of the conditional expectations of $X^2$ and $Y^2$ given that $X^2 + Y^2 = D$.

Q: How do I calculate the conditional expectation of $X^2$ given that $X^2 + Y^2 = D$?

A: To calculate the conditional expectation of $X^2$ given that $X^2 + Y^2 = D$, you can use the formula:

$$E[X^2|X^2 + Y^2 = D] = \frac{\sigma^2(1-\rho^2) + \rho^2 x_0^2}{\sigma^2(1-\rho^2) + x_0^2} x_0^2$$

Q: How do I calculate the conditional expectation of $Y^2$ given that $X^2 + Y^2 = D$?

A: To calculate the conditional expectation of $Y^2$ given that $X^2 + Y^2 = D$, you can use the formula:

$$E[Y^2|X^2 + Y^2 = D] = \frac{\sigma^2(1-\rho^2) + \rho^2 y_0^2}{\sigma^2(1-\rho^2) + y_0^2} y_0^2$$

Q: What is the relationship between the conditional expectation of $X^2Y^2$ and the conditional expectations of $X^2$ and $Y^2$?

A: The conditional expectation of $X^2Y^2$ is the product of the conditional expectations of $X^2$ and $Y^2$.

Q: Can I use the same formulas to calculate the conditional expectation of $X^2Y^2$ for any two Gaussian random variables?

A: No, the formulas used in this article are specific to two independent Gaussian random variables with equal variance. If you have two Gaussian random variables with different variances or a different relationship between them, you will need to use a different approach to calculate the conditional expectation of $X^2Y^2$.

Q: How do I apply the results of this article to real-world problems?

A: The results of this article can be applied to a wide range of real-world problems, including:

• Signal processing: The conditional expectation of $X^2Y^2$ can be used to calculate the power of a signal in a noisy environment.
• Image processing: The conditional expectation of $X^2Y^2$ can be used to calculate the intensity of an image in a noisy environment.
• Finance: The conditional expectation of $X^2Y^2$ can be used to calculate the value of a portfolio in a noisy market.

Conclusion

In this article, we answered some frequently asked questions related to the conditional expectation of a quadratic form of two independent Gaussian random variables, given a condition on their sum of squares. We provided formulas for calculating the conditional expectations of $X^2$ and $Y^2$ and discussed the relationship between the conditional expectation of $X^2Y^2$ and the conditional expectations of $X^2$ and $Y^2$. We also discussed the application of the results of this article to real-world problems.