Let $X$ and $Y$ be i.i.d. $\text{Expo}(λ)$, and $T = X + Y$ .
(a) Find the conditional CDF of $T$ given $X = x$.
(b) Find the conditional PDF $f_{X\mid T}(x\mid t)$, and verify that it is a valid PDF.
Can you help me to verify that the PDF is valid?
I found the following PDF
$P(T \le t\mid X=x)=P(Y \le t-x)=1-e^{-λ(t-x)}\implies$
$f(T \le t \mid X=x)=λe^{-λ(t-x)}$, to verify that it is valid:
$\int_{0}^{\infty} λe^{-λ(t-x)}$ and I compute that the integral is divergent!
The integral is not divergent. \begin{align*}\int\limits_{0}^{\infty}f_{T|X}(t|x)\text{d}t=\int_{x}^{\infty}\lambda e^{-\lambda(t-x)}\text{d}t=\left.-e^{-\lambda(t-x)}\right\vert_{x}^{\infty}=1\end{align*}
The reason is that $t\ge x$.