I have two uncorrelated random variables $X$ and $Y$, both are normally distributed with zero mean with variance of $\sigma^2_X$ and $\sigma^2_Y$ respectively. I would like to know the conditional distribution of $X$ given $Y=-1$.
If we denoted the pdf of $X$ and $Y$ as $f_x(x)$ and $f_y(y)$, then I think that such conditional distribution should be $\frac{f_x(x)f_y(-1)}{f_y(-1)} = f_x(x)$, but I am not very confident about this.
Could someone please help me out?