Joint Distribution of two iid Standard Gaussian variables

Question

I've recently been asked this during an interview and I'm very curious about how to solve questions like these because I tried looking it up with almost no results. Given $X$, $Y$ that are i.i.d. random variables with distributions $N(0,1)$, what are the distributions (mean and variance) of the following:

1. $ X+Y | Y = y $

2. $ X | X+Y=c$

Everywhere I looked seemed to give answers for the distributions of $X|Y$ or something of the like. What do I do with linear transformations of gaussian RV's?

Answer 1

In the first problem,

$X+Y=X+y\sim N(y,\,1)$.

The mean & variance are easy to prove; if it's not obvious to you that the distributions are as claimed, prove it with e.g. MGFs.

Let $p$ denote the joint PDF of $X,\,Y$ in the second problem, so the marginal PDF of $X$ is$$\frac{p(X=x,\,Y=c-x)}{\int_{\Bbb R}p(X=x^\prime,\,Y=c-x^\prime)dx^\prime}=\frac{e^{-x^2+cx-c^2/2}}{\int_{\Bbb R}e^{-x^{\prime2}+cx^\prime-c^2/2}dx^\prime}\propto e^{-(x-c/2)^2},$$i.e. $X\sim N(c/2,\,1/2)$.