Proof of joint Gaussian distribution in a linear equation

Question

Let $X = AY + Z$ where $A$ is constant, $Y$ and $Z$ are independent gaussian rvs. How do you prove that $X$ and $Y$ are jointly gaussian? I know that the sum of two independent gaussians gives a gaussian distribution. But I have no idea where to start to show joint gaussian pdf.

Answer 1

For every linear combination of $X$ and $Y$ : $$ \mu X + \beta Y = (\beta + \mu A) Y + \mu A Z$$ And since $Y$ and $Z$ are independent, the linear combination follows a gaussian distribution. So they are jointly gaussian.