Distribution after adding noise to a Gaussian Distribution

Question

I have a random variable $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$. Now, I add a noise $N \sim \mathcal{N}(0, \sigma_N^2)$ to $X$ to get $Y$ ($Y = X + N$). Thus, $Y \sim \mathcal{N}(\mu_X, \sigma_X^2 + \sigma_N^2)$. Are $Y$ and $X$ bivariate normal now? If so, what would be the density function $f_{Y|X}$?

Answer 1

We know that $$X+N\: | \: X=x \sim x+N \: | \: X=x.$$ If we also know that $X$ and $N$ are independent, then we can infer that $$x+N \: | \: X=x \sim x+N \sim \mathcal{N}(x, \sigma_N^2),$$ which means that $Y|X \sim \mathcal{N}(X,\sigma_N^2)$.

For the part about bivariate normal distribution, you may use that linear transformations of a bivariate normal vector is again bivariate normal. So if $(X,N)$ is bivariate normal (which it is if we assume independence), then $(X,X+N)$ is also bivariate normal.