Answer to Question #242567 in Statistics and Probability for PRS

Question

if the random variable X and Y have the means μ_x=5 μ_y=10 the variances σ²_x=16,σ²_y=9 and ρ=0.3. Obtain the conditional probability distribution of Y given X=2

Answer 1

$\mu_x=5$ $\sigma^2_x=16$

$\mu_y=10$ $\sigma^2_y=9$

The correlation coefficient $\rho=0.3$

We need to determine the conditional probability distribution of $(Y|X=2)$.

Now,

Let $\mu_*$ and $\sigma^2_*$ be the mean and variance of $(Y|X=2)$ respectively. Then, the conditional probability distribution of $(Y|X=2)$ follows a normal distribution with parameters $\mu_*$ and $\sigma^2_*$ and is written as, $(Y|X=2)\sim N(\mu_*,\sigma^2_*)$.

To find the values for the parameters $\mu_*$ and $\sigma^2_*$ , we proceed as follows.

$\mu_*$ is given as,

$\mu_*=\mu_y+\rho(\sigma_y/\sigma_x)*(X-\mu_x)$

$\mu_*=10+0.3(3/4)*(2-5)=9.325$

and $\sigma^2_*$ is given as,

$\sigma^2_*=\sigma^2_y(1-\rho^2)$

$\sigma^2_*=9(1-0.3^2)=9(1-0.09)=9(0.91)=8.19$

Therefore, $(Y|X=2)$, follows a normal distribution with mean, $\mu_*=9.325$ and variance, $\sigma^2_*=8.19$ and can be written as, $(Y|X=2)\sim N(9.325,8.19)$.