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

Asked Jun 09 '22 at 05:53

Active Jun 17 '22 at 12:26

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I have 2 mutually independent random variables:

$s$ is distributed exponentially with parameter $\lambda$: $s\sim F(\cdot|\lambda)$
$\epsilon_x$ 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 $\mathbb{E}(s|s_0=2)$?

edited Jun 17 '22 at 12:26

kjetil b halvorsen

77,844

asked Jun 09 '22 at 05:53

Ludwig Gershwin

You mean $E(s|z_0=2)$? – mpiktas Jun 09 '22 at 06:08
@mpiktas no, $E(s|s_0=2)$. – Ludwig Gershwin Jun 09 '22 at 06:24
@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. – Ludwig Gershwin Jun 09 '22 at 06:40
@Xi'an I'm also very new to probability theory, sorry. I really don't know what to do. – Ludwig Gershwin Jun 09 '22 at 06:53
This is a textbook calculation of conditional densities. – mpiktas Jun 10 '22 at 08:39
@mpiktas if it's that easy, could you please show me how to? – Ludwig Gershwin Jun 10 '22 at 08:57

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

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