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Let $X$ be a random variable, $t \in R, M_X(t) = E(exp(t*i*X))$ the moment-generating function. Proof that $M_X$ is well defined.

I assume I have to proof that if the random variables $X_1$ and $X_2$ have the same distribution, then $E(exp(t*i*X_1)) = E(exp(t*i*X_2))$ should hold. But I don't know how to show it.

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  1. This is normally called the characteristic function.

  2. It wants you to show that $E(\exp{(itX)})$ is finite for any probability distribution. This amounts to $$ \lvert E(e^{itX}) \rvert = \left| \int_{\Omega} e^{itx} dP(x) \right| \leq \int_{\Omega} \lvert e^{itx} \rvert \, dP(x) \leq \int_{\Omega} 1 \, dP(x) = 1. $$

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  • $\begingroup$ Thank you for helping me! One question: How should I have known I had to proof the finiteness? $\endgroup$ – Infinite_28 May 29 '17 at 23:11
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    $\begingroup$ I think this is likely a slightly sloppy use of the term to mean "definition gives a sensible output for every input". Sensible here meaning finite. It's conventional, anyway: one can also speak of a well-defined output (and I almost did two sentences ago, before I caught myself). $\endgroup$ – Chappers May 29 '17 at 23:19
  • $\begingroup$ Okay, thanks a lot! $\endgroup$ – Infinite_28 May 29 '17 at 23:25

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