How to find $\mathbb{E}(V)$ where $V = \frac{100}{2^X}$ and $X\sim \Gamma(3,1)$.

Asked Apr 12 '18 at 08:40

Active Apr 12 '18 at 08:40

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How would I compute this? I tried doing: $$ \mathbb{E}(V) = \mathbb{E}(\mathbb{E}(V|X)) = \mathbb{E} (\mathbb{E}(\frac{100}{2^X}|X=x)) $$ but that seems to not help at all... since doesn't the inner expectation just become $100/2^X$?

asked Apr 12 '18 at 08:40

Mr. Bromwich I

Have you tried computing the corresponding integral? – lcv Apr 12 '18 at 09:02
Hint: If $f$ is the PDF of $X$, then $$ \mathbb E [g(X)] = \int_{-\infty}^\infty g(x)f(x) ,\mathrm d x. $$ – Theoretical Economist Apr 12 '18 at 09:10
I tried but wasn't sure if it's correct.
Since why couldn't I just do $\mathbb{E}(V) = \mathbb{E}(\frac{100}{2^X} )$ instead of usnig the law of iterated expectation? – Mr. Bromwich I Apr 12 '18 at 09:29
From what I understand, $E(\frac{100}{2^X} | X=x) = \frac{100}{2^X}$ since it's like keeping $X$ constant – Mr. Bromwich I Apr 12 '18 at 09:35

How to find $\mathbb{E}(V)$ where $V = \frac{100}{2^X}$ and $X\sim \Gamma(3,1)$.

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