Conditional distribution of $X$ given $X+Y$ is a constant

Question

Let $X$ and $Y$ be independent $\exp(1)$- distributed random variables. Find the conditional distribution of $X$ given that $X+Y = c$. ($c$ is a positive constant).

I have tried to solve this using convolution technique and change of variables but get stuck at the integration part. I think I'm over complicating this so I would love to see different approaches solving this. Thanks.

Answer 1

Using a convolution technique is the right approach.

Since $f_{X,X+Y}(x,c)~ {= ~ f_{X,Y}(x,c-x) \qquad~\,\text{by a Jacobian transform} \\ =f_X(x)f_Y(c-x)\qquad\text{by independence}}$

Therefore, $f_{X\mid X+Y}(x\mid c) ~=~ \dfrac{f_X(x)f_Y(c-x)}{\int_\Bbb R f_X(s)f_Y(c-s)\operatorname d s}$

Which readily simplifies into the pdf of an easily recognised distribution.

Just substitute $f_X(w)=f_Y(w)=\mathsf e^{-w}\mathbf 1_{(0\leqslant w)}$ and pay careful attention to the support.