# How to compute expectation of a exponentially distributed variable given the value of another variable?

I have 2 mutually independent random variables:

1. $$s$$ is distributed exponentially with parameter $$\lambda$$: $$s\sim F(\cdot|\lambda)$$
2. $$\epsilon_x$$ is distributed exponentially with parameter $$\chi$$: $$\epsilon_x\sim F(\cdot|\chi)$$

Define $$x\equiv s-\epsilon_x$$, $$s_0\equiv \max\{x,0\}$$. How do I compute the expectation of $$s$$, given a specific $$s_0$$? For example, How to compute $$\mathbb{E}(s|s_0=2)$$?

• You mean $E(s|z_0=2)$? Commented Jun 9, 2022 at 6:08
• @mpiktas no, $E(s|s_0=2)$. Commented Jun 9, 2022 at 6:24
• @Xi'an This is just the first step of a research question. You can ignore $b$ and $z_0$. Given $s_0=2$, we have $s-\epsilon_x=2$, then...? I'm sorry I'm very new to mathematical statistics. Commented Jun 9, 2022 at 6:40
• @Xi'an I'm also very new to probability theory, sorry. I really don't know what to do. Commented Jun 9, 2022 at 6:53
• This is a textbook calculation of conditional densities. Commented Jun 10, 2022 at 8:39