Suppose we have two random variables, $X \sim \mathcal{N}(\mu_1, \sigma^2)$ and $Y \sim \mathcal{N}(\mu_2, \sigma^2)$ and $(X, Y)$ has a bivariate normal distribution. In addition, the correlation coefficient between $X$ and $Y$ is $t$.
I am trying to compute the joint PDF of $(X, Y + t X)$, but I am not sure how to start.
I think that I need to find the PDF of $Y + t X$ first. Any ideas are more than welcome.
Edit: Added the correlation between $X$ and $Y$.