Problem 58 - Applied Statistics and Probability for Engineers

Question

Suppose that \(X\) and \(Y\) have a bivariate normal distribution with joint probability density function \(f_{X Y}\left(x, y ; \sigma_{X}, \sigma_{Y}, \mu_{X}, \mu_{Y}, \rho\right)\) (a) Show that the conditional distribution of \(Y\) given that \(X=x\) is normal. (b) \(\quad\) Determine \(E(Y \mid X=x)\). (c) \(\quad\) Determine \(V(Y \mid X=x)\).

Step-by-Step Solution

Answer

(a) The conditional distribution of \(Y\) given \(X=x\) is normal. (b) \(E(Y|X=x) = \mu_Y + \rho \frac{\sigma_Y}{\sigma_X} (x - \mu_X)\). (c) \(V(Y|X=x) = \sigma_Y^2 (1 - \rho^2)\).

Step 1: Understand the Characteristics of Bivariate Normal Distribution

Given that the random variables \(X\) and \(Y\) have a bivariate normal distribution, we can express the joint probability density function as: \[f_{X Y}(x, y ; \sigma_{X}, \sigma_{Y}, \mu_{X}, \mu_{Y}, \rho) = \frac{1}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1-\rho^2}} \exp \left(-\frac{1}{2(1-\rho^2)} \left[ \frac{(x-\mu_X)^2}{\sigma_{X}^2} - \frac{2\rho(x-\mu_X)(y-\mu_Y)}{\sigma_X \sigma_Y} + \frac{(y-\mu_Y)^2}{\sigma_Y^2} \right] \right)\]Where \(\mu_X\) and \(\mu_Y\) are the means, \(\sigma_X\) and \(\sigma_Y\) are the standard deviations, and \(\rho\) is the correlation coefficient.

Step 2: Determine the Formulas for Conditional Distribution

The conditional distribution of \(Y\) given \(X = x\) for variables with a bivariate normal distribution is also normal, and can be described by:\[ Y | X = x \sim N(\mu_{Y|X=x}, \sigma^2_{Y|X=x}) \]where \(\mu_{Y|X=x}\) and \(\sigma^2_{Y|X=x}\) are the conditional mean and variance respectively.

Step 3: Calculate the Conditional Mean \(E(Y \mid X=x)\)

The conditional mean \(\mu_{Y|X=x}\) of \(Y\) given \(X = x\) is determined by the formula:\[ \mu_{Y|X=x} = \mu_Y + \rho \frac{\sigma_Y}{\sigma_X} (x - \mu_X) \]

Step 4: Calculate the Conditional Variance \(V(Y \mid X=x)\)

The conditional variance \(\sigma^2_{Y|X=x}\) of \(Y\) given \(X = x\) is given by:\[ \sigma^2_{Y|X=x} = \sigma_Y^2 (1 - \rho^2) \]

Step 5: Conclusion and Verification

The conditional distribution \(Y|X=x\) is indeed normal as shown by the mean \(\mu_{Y|X=x}\) and variance \(\sigma^2_{Y|X=x}\). With these parameters defined, any conditional expectation and variance calculated aligns with the properties of a normal distribution.

Key Concepts

Conditional Distribution

In a bivariate normal distribution, the conditional distribution is a key concept that describes how one variable behaves when another variable is held at a certain value. Consider the random variables, \(X\) and \(Y\), which are distributed together as a bivariate normal distribution. When we look at how \(Y\) behaves given a specific \(X = x\), it creates a new distribution for \(Y\).
The conditional distribution of \(Y\) given \(X = x\) is also a normal distribution. This is expressed as \(Y | X = x \sim N(\mu_{Y|X=x}, \sigma^2_{Y|X=x})\).
This property of normality tells us a lot. It means that the behavior of \(Y\), given some value of \(X\), can be modeled and predicted using a normal distribution with specific parameters - the conditional mean and conditional variance - which are crucial for our understanding.

Conditional Mean

The conditional mean is a measure of the expected value of \(Y\) when \(X\) is at a particular point \(x\). In a bivariate normal distribution, this conditional mean provides insight into how changes in \(X\) affect \(Y\)'s expected outcome.
The formula for conditional mean is given by:
\[ \mu_{Y|X=x} = \mu_Y + \rho \frac{\sigma_Y}{\sigma_X} (x - \mu_X) \]
Here’s a breakdown of the components:

• \(\mu_Y\) is the mean of \(Y\).
• \(\rho\) is the correlation coefficient, indicating the strength and direction of the relationship between \(X\) and \(Y\).
• \(\sigma_Y\) and \(\sigma_X\) are the standard deviations of \(Y\) and \(X\) respectively.
• \(x\) is the specific value of \(X\) we are considering.
• \(\mu_X\) is the mean of \(X\).

The conditional mean adjusts the average of \(Y\) based on how far \(X\) is from its mean, scaled by their correlation and variability.

Conditional Variance

Conditional variance describes the variability of \(Y\) given that \(X\) is held at a certain value \(x\). It tells us how spread out \(Y\) is likely to be around the conditional mean \(\mu_{Y|X=x}\).
For a bivariate normal distribution, the conditional variance of \(Y\) given \(X = x\) can be determined using the formula:
\[ \sigma^2_{Y|X=x} = \sigma_Y^2 (1 - \rho^2) \]
This expression involves:

• \(\sigma_Y^2\), the variance of \(Y\) without any conditioning.
• \(\rho\), the correlation coefficient showing the relationship strength between \(X\) and \(Y\). If \(\rho\) is strong, \(\sigma^2_{Y|X=x}\) gets reduced, reflecting that a lot of variability is shared with \(X\).

Thus, conditional variance provides insight into how much uncertainty (or spread) there is in \(Y\), once \(X\) has been accounted for.

Correlation Coefficient

The correlation coefficient, represented by \(\rho\), is a statistical measure that describes the strength and direction of a linear relationship between two random variables \(X\) and \(Y\). It's a key part of the bivariate normal distribution.
When examining the conditional distribution and its properties, the correlation coefficient informs us about:

• The extent to which \(Y\) changes as \(X\) changes. A positive \(\rho\) indicates that the variables tend to move in the same direction, while a negative \(\rho\) shows an inverse relationship.
• The degree to which the changes in \(X\) can impact changes in \(Y\). A high absolute value of \(\rho\) (close to 1 or -1) shows a strong relationship.
• It shapes how the conditional mean and variance of \(Y\) given \(X=x\) are calculated, influencing their values directly.

The beauty of \(\rho\) lies in its ability to succinctly encapsulate the relationship's essence between \(X\) and \(Y\) in a single number, vital for understanding phenomena through the lens of bivariate normal distribution.