Let $X_1$, $X_2$, $X_3$ be a random sample from a discrete distribution with probability funciton
$p(0)= 1/3$
$p(1) = 2/3$
Calculate moment generating function, $M(t)$, of $Y=$$X_1$$X_2$$X_3$
My Work
$M_x(t) = \frac{1}{3} + \frac{2}{3}e^t$
then $E[e^{t(X_1X_2X_3)}]$
$=E[e^{tX_1}+e^{tX_2}+e^{tX_3}]$
$= E[e^{tX_1}]+E[e^{tX_2}]+E[e^{tX_3}]$
$=3(\frac{1}{3} + \frac{2}{3}e^t)$
$=1+2e^t$
However, $M_x(0)=3\neq1$, so this must be wrong, but why?