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I'm trying to answer a question about expectations. Let's say I have a random variable, $X$, which is distributed $Expo(\lambda)$. Now assume that $\lambda$ is also an R.V. and is distributed $Expo(5)$. How do I find $E(X)$? Assume that I don't need to do this analytically, I can use the integrate command in R, for instance. I believe I should be using Adam's Law, but I'm not sure. Thank you!

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    $\begingroup$ This sounds like a self-study question. Can you explain where you have problem with using Adam's Law? $\endgroup$ – Xi'an Feb 12 '18 at 11:34
  • $\begingroup$ I understand the Adam's Law part of it I think (based on Siong's answer below). I'm unsure of how to actually integrate it in R. $\endgroup$ – lish Feb 12 '18 at 16:24
  • $\begingroup$ Why would you want to find a numerical approximation when you can get the exact answer? $\endgroup$ – Xi'an Feb 12 '18 at 17:29
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\begin{align} E[X] &= \int_0^\infty E[X|\lambda]f(\lambda) \, d\lambda \\ &= \int_0^\infty E[X|\lambda](5)\exp(-5\lambda) \, d\lambda \end{align}

Hopefully you can take it from here.

Hint: it is something huge.

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