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Given a gamma distribution with shape $\alpha=2$ and rate $\lambda=10$, I was first asked to find an expression for $\Bbb E[X^k] \ \forall \ k \in \Bbb N$. Directly computing this, I got $$\Bbb E[X^k]=\frac{\Gamma(k+\alpha)}{\lambda^k\Gamma(\alpha)}.$$ Now I am asked to find an expression for $\Bbb E[X^{-2}]$, which given my parameters, would result in the above expression including $\Gamma(0)$ which is not defined. How should I go about this calculation, and is simply using the moment generating function the only way to do so? I understand that the MGF can only be used for nonnegative k, so I don't think that will work.

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  • $\begingroup$ $EX^{-2}=\infty$. $\endgroup$ – Kabo Murphy May 23 at 5:20
  • $\begingroup$ How did you arrive at this answer? $\endgroup$ – mathenthusiast May 23 at 5:25
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$EX^{-2}=100\int_0^{\infty} \frac 1 x e^{-10x}\, dx$. Since $e^{-10x} \to 1$ as $x \to 0$ and $\int_0^{1} \frac 1 x \, dx=\infty$ it follows that $EX^{-2}=\infty$.

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