We have a set of Random Variables $Y_i$ which takes the value $\alpha$ with probability $(1-p)$ and takes the value $1-\alpha$ with a probability of $p$.
We have been tasked with finding the Moment Generating Function (and the Cumulant Generating Function) of the sample mean of $Y_i$ (ie $S_N=\frac{1}{N}\sum^{N}_{i=0}Y_i$)
Knowing that I can simply use the formula:
$M_{S_N}(t)=[M_{Y_i}(\frac{t}{N})]^N$
I just need to find the MGF of $Y_i$. However, I am having issues obtaining it's pdf and trying to get it into a nice form. I have tried defining it as:
$P(Y_i=y)=(1-p)\delta_{y,\alpha} + p\delta_{y,(1-\alpha)}$
But still can't get the pdf into a usable form.
Why is this difficult? You simply apply the definition:
$$M_{Y_i}(t) = \operatorname{E}[e^{tY_i}] = e^{t\alpha} \Pr[Y_i = \alpha] + e^{t(1-\alpha)}\Pr[Y_i = 1-\alpha] = e^{t\alpha} (1-p) + e^{t(1-\alpha)}p.$$