Covariance of two standard normal random variables

Question

Let $X$ follow the standard normal distribution $N(0,1)$

Let $a>0$

Let $Y=X$ if $|X|<a$

$ $ $ $ $ $ $ $ $ $ $ $ $Y=-X$ if $|X|\geq a$

Then, it is easily shown that Y follows the standard normal distribution $N(0,1)$

1. What is $Cov(X,Y)$?

2. Is the random vector $(X,Y)$ a multivariate normal?

Firstly, since $E(X)=E(Y)=0$, $Cov(X,Y)=E(XY)$

I think $E(XY)=E(X^{2}1_{|X|<a}) - E(X^{2}1_{|X|\geq a})$

how do I proceed to solve this problem?

Secondly, I think the random vector is not multivariate normal since if it were, it would be true that $X+Y$ is normal but I think it is not

Answer 1

You're most of the way there. It only remains to calculate $E(X^21_{\mid X\mid<a})$.

This can be written as an integral $2\int_0^a x^2 f_X(x)dx$ where $f_X$ is the density of $X$. To evaluate this, note that $f_X'(x)=-xf_X(x)$, so the integral is $-2\int_0^a xf'_X(x)dx$. Now integrate by parts, and you get an explicit formula involving the CDF of X. The second term $E(X^21_{\mid X\mid\ge a})$ doesn't require any additional calculations because it is equal to $E(X^2)-E(X^21_{\mid X \mid<a})$.