# Law of total variance and covariance given X and Y are normal

I have a problem which asks me to find $$\Bbb E[Y]$$ and $$Var(Y)$$ given that $$Y\text{~}Normal(x,1)$$ conditional on $$X=x$$. $$X$$ is standard normal. So I have worked out that $$\Bbb E[Y]=0$$ using the law of iterated expectations; now I want to work out the variance. I figure I need to use the law of total expectation, which states that $$Var(Y)=\Bbb E[Var(Y|X)]+Var(\Bbb E[Y|X])$$

I have worked out that $$\Bbb E[Var(Y|X)]=1$$, so all I need is $$Var(\Bbb E[Y|X])$$, which is given by $$Var(\Bbb E[Y|X])=\Bbb E[\Bbb E[Y|X]^2]-(\Bbb E[\Bbb E[Y|X]])^2$$ I can further work out that $$(\Bbb E[\Bbb E[Y|X]])^2=\Bbb E[Y]^2=0$$; what I cannot figure out is $$\Bbb E[\Bbb E[Y|X]^2]$$.

I understand that $$\Bbb E[Y|X=x]=x$$, so I assume that means $$\Bbb E[Y|X]=X$$, so does that mean that $$\Bbb E[Y|X]^2=X^2$$? If so, I take it that $$\Bbb E[\Bbb E[Y|X]^2]=0$$, because $$X$$ is standard normal. Am I correct? If not, where is my error?

I am also wondering how to work out $$\text{Covariance}(X,Y)$$ once the variance is calculated. Namely, what will the distribution $$XY$$ be? Because $$\Bbb E[XY]$$ is needed. Is it bivariate normal? Any help is greatly appreciated.

• How did you find $\mathbb EY=0$? We have $\mathbb E[Y\mid X]=X$ so $\mathbb EY=\mathbb E[\mathbb E[Y\mid X]]=\mathbb EX$. What exactly do you know about $X$? – drhab May 23 '19 at 13:29
• Sorry, should have mentioned $X$ is standard normal. Will edit that in. – mathenthusiast May 23 '19 at 13:31

I think you can do it with:$$Y=X+U$$where $$U$$ has standard normal distribution and $$X$$ and $$U$$ are independent. In that case under condition $$X=x$$ random variable $$Y$$ has normal distribution with parameters $$\mu=x$$ and $$\sigma^2=1$$ (as is requested).
If moreover $$X$$ has standard normal distribution then it is immediate that $$Y$$ has normal distribution with parameters $$\mu_Y=0$$ and $$\sigma_Y^2=2$$.