Chapter 7, Problem 7.42TE

Let X be a normal random variable with parameters $\mu =0$ and ${\sigma }^2=1$, and let I, independent of X, be such that $P\left\{I=1\right\}=\frac{1}{2}=P\left\{I=0\right\}$. Now define Y by $Y=\{\begin{array}{cc}X& \text{if }I=1\\ -X& \text{if }I=0\end{array}$

In words. Y is equally likely to equal either X or -X.

a. Are X and Y independent?

b. Are I and Y independent?

c. Show that Y is normal with mean 0 and variance 1.

d. Show that $Cov\left(X,Y\right)=0$.