The input $X$ to a binary communication channel assumes the value $+1$ or $-1$ with probability $\frac{1}{3}$ and $\frac{2}{3}$ respectively. The output of $Y$ of the AWGN channel is given by $Y=X+N$ where $N$ is zero mean Gaussian noise with variance $=1.$
Find the conditional pdf of $Y$ given $X=+1$.
So I started working on it:
$P_X(\{1\})=\frac{1}{3}$
$P_X(\{-1\})=\frac{2}{3}$
I am not sure how to deal with N.