Moment generating function for independent random variables

Question

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.

Answer 1

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.