The joint density of $X$ and $Y$ is given by : $f(x,y)=\frac{1}{\sqrt{2 \pi}}e^{-y}e^{-\frac{(x-y)^2}{2}}$. 
The joint density of $X$ and $Y$ is given by
  $$f\left(x,y\right)=\frac{1}{\sqrt{2\pi}}e^{-y}e^{-\frac{\left(x-y
\right)^{2}}{2}}$$
  Here $0<y<\infty$ and $-\infty<x<\infty$. Compute the individual MGFs of $X$ and $Y$.

I could find the marginal distribution of $Y$ which is ${\rm Exp}\left(1\right)$, so I can find its MGF. But I cannot find the marginal distribution of $X$. Please help!

Assuming "MGF" means "moment generating function":
In this set-up, $Y$ is exponential and conditional on $Y$, the rv. $X$ is $N(Y,1)$.  That is, $X$ can be represented as $Y+Z$, where $Z\sim N(0,1)$ is independent of $Y$.  So the MGF of $X$ is $E\exp (uX) = E\exp(uY+uZ) = E\exp(uY)\exp(uZ)=\frac 1{1-u}\exp(u^2/2)$.  The MGF of $Y$ is $E\exp(vY)=1/(1-v)$.
