if the random variable X and Y have the means μx=5 μy=10 the variances σ2x=16,σ2y=9 and ρ=0.3. Obtain the conditional probability distribution of Y given X=2
"\\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)".
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