What variable is the conditional probability function a function of?

Question

I've got the following excerpt from my notes in my statistics course:

In the second line, the professor put $x=1$ in the Poisson distribution equation. My question, how do I know that the conditional distribution function is a function of $X_2$, and not $X_1$?

Answer 1

You are given that $$ P(X_2 = j | X_1 = k) = e^{-k} \frac{k^j}{j!} $$ for any nonnegative integers $j, k$. Now just plug in $j=1$ to obtain $$ P(X_2 = 1 | X_1 = k) = e^{-k} \frac{k^1}{1!}. $$