Based on the question here, I am confused about computing the conditional density of a random variable conditioned on a sum involving it.
Let the setup be $X$ and $Y$ are independent exponentials with rates $\lambda_X$ and $\lambda_Y$. Let $T=X+Y$. How to compute the conditional density $f_{X \mid T}(x \mid t)$?
I get $$ f_{X \mid T}(x\mid t) = -(\lambda_X - \lambda_Y) e^{-\lambda_X x} e^{-\lambda_Y (t-x)} / (e^{-\lambda_Xx} - e^{-\lambda_Yy}) $$ by chasing definitions of densities. Is this right?
I find densities unintuitive and would very much appreciate an argument using conditional expectations if possible!
Hints: Consider $P(X \leq x |T)$. This can be written as $g(T)$ for some measurable function $g$ and $g$ is computed using the equation $$\int_{X+Y \leq t} g(X+Y)dP=\int_{X+Y \leq t} XdP.$$ for all $t$. The right side can be computed using the joint density of $X$ and $Y$ and the left side is $\int_{-\infty}^{t} g(u)\phi (u)du$ where $\phi$ is the density of $X+Y$. Differentiating this (w.r.t. $t$) we can find $g$ as a function of $x$ and differentiation (w.r.t. $x$) gives the density of $X$ given $T$.