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.
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).  this is my idea:  $$f_{X \mid X+Y}(x\mid c) = \frac{f_{X,X+Y}(x,c)}{f_{X+Y}(c) } $$  if this is the right definition then i might calculate the righthand side to obtain the distribution but Iam unsure
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