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Let $X_1, X_2, \ldots, X_n$ be independent random variables that all have the same distribution, let $N$ be an independent non-negative integer valued random variable, and let $S_N := X_1 + X_2 + \cdots + X_N$. Find an expression for the moment generating function of $S_N$.

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    $\begingroup$ I have retyped your mathematical expressions. Please click "edit" can take some time to learn how to do what I did. $\endgroup$ – Fly by Night Jul 20 '13 at 22:05
  • $\begingroup$ The answer will depend upon the distribution of $X_1$. What sort of answer do you want? For example, do you want something that depends upon the moment generating function of $X_1$? $\endgroup$ – Greg Martin Jul 20 '13 at 23:07
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    $\begingroup$ This may help: math.stackexchange.com/questions/338129/… $\endgroup$ – user940 Jul 20 '13 at 23:56
  • $\begingroup$ All I know is that it is i.i.d random variables. There is nothing else given in the question. This was on my test I had today..and thank you for the link! $\endgroup$ – oyth94 Jul 21 '13 at 1:39
  • $\begingroup$ I am also confused with the part where it mentions N. is N another random variable itself or the number of random variables..? $\endgroup$ – oyth94 Jul 21 '13 at 3:56
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Hints:

  • For every $t$ in $(0,1)$, $E[t^{S_N}]=\sum\limits_{n=0}^{+\infty}P[N=n]\cdot E[t^{S_n}]$. (Which hypothesis is this identity based on? which definition of $S_0$ is required for this step to hold?)
  • For every $n\geqslant0$, $E[t^{S_n}]=\left(E[t^{X_1}]\right)^n$. (Which hypothesis is this identity based on? which definition of $S_0$ is required for this step to hold?)
  • Hence, for every $t$ in $(0,1)$, $E[t^{S_N}]=$ $______$ where $\varphi(t)=E[t^{X_1}]$ and $\psi(t)=E[t^N]$.
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  • $\begingroup$ Sorry I'm not following .. $\endgroup$ – oyth94 Jul 25 '13 at 1:52
  • $\begingroup$ Yes?? $ $ $ $ $ $ $\endgroup$ – Did Jul 25 '13 at 6:57
  • $\begingroup$ If $X_1,X_2,\cdots,X_n$ be independent random variables from the same continuous distribution and $P=\prod_{i=1}^n X_i$. Is the mgf of $P$ $$M_P(t)=\mathbf{E}\left[e^{tP}\right]=\underbrace{\int_{-\infty}^\infty\cdots \int_{-\infty}^\infty}_{n\text{ times}}e^{t\prod_{i=1}^n X_i}\prod_{i=1}^n f_{X_i}(x_i)\,dx_1\cdots dx_n\,?$$Does it also hold for the different distributions? $\endgroup$ – Venus Jun 17 '15 at 13:44
  • $\begingroup$ @Venus Is this related to the question on this page? $\endgroup$ – Did Jun 17 '15 at 14:08
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    $\begingroup$ @Venus Yes it is (correct). Note that your last comment is rather bizarre: "independent random variables from the same continuous distribution" is the case you said in your first comment that you could handle. $\endgroup$ – Did Jun 21 '15 at 13:15

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