If $X,Y\sim N(0, σ^2)$ are independent, what is the conditional distribution of $X$ given $X+Y=12$?

Question

If $X,Y\sim N(0, σ^2)$ are independent, find the conditional distribution of $X$ given $X+Y=12$.

The answer is $X |_{X+Y =12} \sim N(6, σ^2/2)$. It seems not that hard but I couldn't get it right.

Thanks.

Answer 1

You must evaluate: $$f_{X\mid X+Y}(x\mid 12)=\dfrac{f_X(x)~f_Y(12-x)}{f_{X+Y}(12)}$$

Now since $X,Y$ are normally distributed, the numerator is easy: $$f_X(x)~f_Y(12-x)=(2\pi\sigma^2)^{-1/2}\exp(-x^2/(2\sigma^2))\cdot (2\pi\sigma^2)^{-1/2}\exp(-(12-x)^2/(2\sigma^2))$$

But what is the probability density function of $X+Y$ , the sum of two independent and normally distributed random variables?