I have this problem:
Let the random vector $\binom{X}{Y}\sim
N_{2}(\binom{0}{0},\bigl(\begin{smallmatrix} 1 & \rho\\ \rho & 1 \end{smallmatrix}\bigr))$1) Find the distribution of $Z=X+Y$.
2) Find the distribution of $W=X^2$, the mean and the variance.
3) Calculate $Cov(X,W)$ and $Cov(Z,W)$.
It's the first time that I have found myself analysing this type of function. What the covariance matrix involves? How do I manage it? Thanks in advance for any clarification!
Best Answer
It is given that $X$ and $Y$ are jointly normal, $EX=EY=0$, $EX^{2}=EY^{2}=1$ and $cov(X,Y)=\rho$ which gives $EXY=\rho$.
$X+Y$ is normal with mean $0$ and its variance is $E(X+Y)^{2}=EX^{2}+EY^{2}+2EXY=1+1+2\rho$.
$P(W \leq w)=P-\sqrt w \leq X \leq \sqrt w)-\int_{-\sqrt w} ^{\sqrt w} f(x)dx$ where $f$ is the standard normal density.
$cov(X,W)=EX^{3}-EXEX^{2}=EX^{3}=0$ by symmetry of standard normal distribution. I will leave the calculation of $cov (Z,W)$ to you.