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Suppose that $X_1, X_2$, and $X_3$ are identically distributed independent discrete random variables with probabilities given by...

  • $1/3$ if $x=0$
  • $2/3$ if $x=1$
  • $0$ else

Find the moment generating function of $Y=X_1X_2X_3$

So far, I have the following

$\sum \exp(Yt)\cdot p(Y)$

but I have no idea where to go from there.

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Hint: First find the distribution of $Y$. It is not hard, since $Y=1$ if all the $X_i$ are $1$, and $Y=0$ otherwise.

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  • $\begingroup$ Yea, they are independent, I missed that. $\endgroup$ – Demetri Pananos Dec 12 '13 at 23:10

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