# Conditional distribution involving two normal distributions

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?

• What does it say about the independence of two random variables if they are uncorrelated? – Tony Hellmuth May 28 '18 at 6:35
• @TonyHellmuth I don't think uncorrelation implies independence, but I could not think of solving this without assuming independence – James May 28 '18 at 6:37
• Now, lets think about this from the perspective they happen to be Normal random variables. Are they then not independent? Why or why not? – Tony Hellmuth May 28 '18 at 6:39
• @TonyHellmuth I have definitely come across a statement saying that if the joint distribution of two uncorrelated random variables are normal, then the two are independent, but I am not sure about the normality for the joint distribution of X and Y. – James May 28 '18 at 6:47
• I think the argument is solid that it should follow since the CDF show X and Y are independent, we have $f_{X|Y}(x,-1)=f_{X}(x)$ as we have no more information on how X and Y are related. You can read up here: probability.ca/jeff/teaching/uncornor.html In fact I think all we cannot say is that the joint distribution is normal. It may be that $f_{X,Y}(x,y)$ is a mixture of some sort. – Tony Hellmuth May 28 '18 at 7:14