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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 $\mathbb{E}(s|s_0=2)$?

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  • $\begingroup$ You mean $E(s|z_0=2)$? $\endgroup$
    – mpiktas
    Commented Jun 9, 2022 at 6:08
  • $\begingroup$ @mpiktas no, $E(s|s_0=2)$. $\endgroup$ Commented Jun 9, 2022 at 6:24
  • $\begingroup$ @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. $\endgroup$ Commented Jun 9, 2022 at 6:40
  • $\begingroup$ @Xi'an I'm also very new to probability theory, sorry. I really don't know what to do. $\endgroup$ Commented Jun 9, 2022 at 6:53
  • $\begingroup$ This is a textbook calculation of conditional densities. $\endgroup$
    – mpiktas
    Commented Jun 10, 2022 at 8:39

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