Question

Let X be a standard normal random variable. Let Y be a continuous random variable such that

fY|X(y|x)=1√2π*exp(−((y+2x)^2)/2)
1. Find E[Y|X=x] (as a function of x, in standard notation) and E[Y].

E[Y|X=x]= unanswered

E[Y]= unanswered

2. Compute Cov(X,Y).
Cov(X,Y)= unanswered

3. The conditional PDF of X given Y=y is of the form

α(y)exp{−quadratic(x,y)}

By examining the coefficients of the quadratic function in the exponent, find E[X∣Y=y] and Var(X∣Y=y).

E[X∣Y=y]= unanswered

Var(X∣Y=y)= unanswered