Question

Q1. (20 points total) Let X ~ ŅĢii'ơf) and Y ~ ŅĢi2, σ ). Answer the following questions (10 points) If ρ-Corr(X Y) normal distribution with meain 1. 0, show that the conditional distribution of Y given X-x is a and variance Var(Y x)-σ (1-ρ2). That is, prove that the conditional PDF of Y given X-X 1S 2. (5 points) The conditional inean of Y given X = x in (1) can be reexpressed as where Ao-μ2-ρ and β-ρσ2. Interpret the meaning of β1 based on this expression (2) 3, (5 points) The conditional inean of Y given X = x in (1) can be reexpressed as σ2 ƠI Interpret the meaning of p based on this expression (3)

Answer 1

$1.~The~Conditional~distribution~of~Y~given~X=x~is\\ f(y|x)=\frac{f(x,y)}{g(x)},~where~marginal~distribution~of~X~is~(\mu_1,\sigma_1^2)\\ =\frac{\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\exp\left(-\frac{1}{2(1-\rho^2)}\left[\left(\frac{x-\mu_1}{\sigma_1} \right )^2-2\rho\left(\frac{x-\mu_1}{\sigma_1} \right )\left(\frac{y-\mu_2}{\sigma_2} \right )&plus;\left(\frac{y-\mu_2}{\sigma_2} \right )^2\right ] \right )}{\frac{1}{\sqrt{2\pi}\sigma_1}\exp\left(-\frac{1}{2}\left(\frac{x-\mu_1}{\sigma_1} \right )^2 \right )}\\ =\frac{1}{\sqrt{2\pi}\sigma_2\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)\sigma_2^2}\left(y-\left(\mu_2&plus;\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1)\right) \right )^2 \right ],\\~-\infty<y<\infty\\$

$Since~\frac{1}{2\pi\sigma_1\sigma_2\sqrt{1-\rho^2}}\exp\left(-\frac{1}{2(1-\rho^2)}\left[\left(\frac{x-\mu_1}{\sigma_1} \right )^2-2\rho\left(\frac{x-\mu_1}{\sigma_1} \right )\left(\frac{y-\mu_2}{\sigma_2} \right )&plus;\left(\frac{y-\mu_2}{\sigma_2} \right )^2\right ] \right )\\ =\frac{1}{\sqrt{2\pi}\sigma_1}\exp\left(-\frac{1}{2}\left(\frac{x-\mu_1}{\sigma_1} \right )^2\right)\times\\ \frac{1}{\sqrt{2\pi}\sigma_2\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)\sigma_2^2}\left(y-\left(\mu_2&plus;\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1)\right) \right )^2 \right ]\\ Hence~Y|x~follows~normal~distribution~with~mean=E(Y|x)\\ =\mu_2&plus;\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1)~ and~variance=Var(Y|x)=\sigma_2^2(1-\rho^2).\\ 2.~E(Y|x)=\mu_2&plus;\rho\frac{\sigma_2}{\sigma_1}(x-\mu_1)=\left(\mu_2-\rho\frac{\sigma_2}{\sigma_1}\times \mu_1 \right )&plus;\rho\frac{\sigma_2}{\sigma_1}\times x\\ =\beta_0&plus;\beta_1x\\$

$where,~\beta_0=\mu_2-\rho\frac{\sigma_2}{\sigma_1}\times \mu_1,~\beta_1=\rho\frac{\sigma_2}{\sigma_1}.\\ \beta_1~means~if~we~increase~one~unit~in~x~then~y~is~increased~by~\beta_1~unit.\\ 3.~E\left(\frac{Y-\mu_2}{\sigma_2}|x \right )=\frac{1}{\sigma_2}E(Y|x)-\frac{\mu_2}{\sigma_2}=\frac{1}{\sigma_2}\left(\mu_2&plus;\rho\frac{\sigma_2}{\sigma_1} (x-\mu_1) \right )-\frac{\mu_2}{\sigma_2}\\ =\rho\left(\frac{x-\mu_1}{\sigma_1} \right )$

$~If~we~increase~one~unit~in~z-score~of~X~then~expected~z~score~of~Y~is~increased~by~\rho.$