Given $X,Y \sim N(0,1)$, X,Y follows a joint normal distribution and $corr(X,Y) = \frac{1}{2}$.
Evaluate $E[e^X|Y=1]$.
I'm not sure where to start on this one. I did a change of variable $Z = e^X$ and use the definition of conditional expectaion: $E[Z|Y=1] = \sum_{z} zp_{Z|Y}(z|y)$ but it isn't helpful.
Since the law of a Gaussian vector of mean zero is determined by its covariance matrix, you can write $X=Y/2+t Z$ where $Y,Z$ are i.i.d. $N(0,1)$ variables and $t=\frac{\sqrt 3}{2}$ (In the sense that this pair $(X,Y)$ has the same joint law as the given variables.) Therefore $$E[e^X|Y=1]=e^{1/2} E[e^{tZ}]= e^{1/2} e^{t^2/2}= e^{5/4} \,$$
https://en.wikipedia.org/wiki/Normal_distribution#Moment_and_cumulant_generating_functions