When I read maths chapter, I found following conditional probability expression in a part of intermediate step and then directly gave the answer \begin{equation} I=\Pr\{x\leq y|x=t\} \end{equation} Here $x$ and $y$ are i.i.d exponential distributions.
Can someone please write $I$ by using CDF/PDF of $x$ and $y$?
Substituting $x=t$ then $$ \begin{align} I & =P(t \le y )\\ &=1-P(y \le t) \end{align} $$
($P(y \le t)$ is the CDF of $y$.)
Given $x$ is a random variable you may be interested in $P(y \le x)$ or equivalently $P(y - x \le 0)$. This concerns the difference of 2 i.i.d. exponential variates. The PDF of the difference of 2 exponential i.i.d. random variables has already been discussed here
