I have a random variable $X$ with density function \begin{equation} f(x)=Nx^{-\alpha}e^{-x}\mathbb{1}_{[1,\infty)}(x) \end{equation} To continue with the main part of the exercise I am working on I need to compute the moment generating function of this random variable. So I note that \begin{equation} M_{X}(t)=\mathbb{E}[e^{tX}]=\int_{1}^{\infty}e^{tx}Nx^{-\alpha}e^{-x}dx \end{equation} If I for example use Wolfram Alpha it becomes clear to me that this integral does not exist and when applying integration by parts it becomes a huge mess for me. Does anyone know if I am making a mistake or how to compute this integral?
2026-03-31 19:08:18.1774984098
Moment generating function problem with integral
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