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I've been having difficulties with the following question:

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I feel this is a very simple problem that I'm overthinking, but I've been stumped on this question for a while. I've tried to simplify it and turn the integral into the gamma function, but I can't figure out how to do it! x is only present in the exp() function, so I can't understand how to transform it into something else. I would appreciate any help directing me to the right way to do this question.

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1 Answer 1

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You're trying to use $\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} \,\mathrm d x$, right? Well, not only is there not an $x^{t-1}$, the exponent has that annoying $q$ in it. Try getting rid of that problem first and then worry about the other one.

More explicit hint, click to reveal:

Use a change of variables to $u = \frac{\lvert x \rvert^q}{2 \sigma^2}$. Careful with the limits of integration when you do.

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