Two random variables where X ~ N(0,1) and Y|X=x ~ N(x,1) (Y is conditionally normal distribution given X=x).
I would to find the marginal distribution of Y.
I try to formulate f(x,y) = f(y|x)*f(x) and then I could do integration with respect to X.
However, I get stuck about how to do the integral with e^[(-1/2)*(y^2 - 2xy + 2x^2)] with respect to x.
Any hint would be appreciated. Thanks.