I've came across this confusion while reading this question:
The time until Sam receives his first email is exponentially distributed with parameter $4$.
The time between the first email and the second email, given that the first email has arrived in time $y$, is exponentially distributed with expectation of $y$.
So, by reading through the question, I wrote on the side $Y$- The times until the first email arrives $\Longrightarrow Y\sim Exp(4)$.
And $X$- time between both emails.
But was confused of how to write the second sentence, I know that $X|Y=y$ should be distributed with parameter $\lambda=\frac{1}{y}$,
but how should it be written?
$X|Y\sim Exp(\frac{1}{y})$ or $X|Y=y \sim Exp(\frac{1}{y})$.
My intuition leans toward the second choice, but I know that $X|Y=y$ is a function of $y$, say $g(y)$, and that is not a random variable, so I'm not sure if I could write it like that.
Thanks in advance!