Moment generating function of product of Rademacher and Bernoulli

Question

We have the following random variables:

$$A_i\sim Bernoulli(a)$$

$$B_i\sim Rademacher\left(b\right)$$

$$C_i\sim Rademacher(c)$$

where $a,b,c\in [0,1]$ are constants for $i=1,\ldots, n$.

All the random variables above are mutually independent.

I'm trying to find the moment generating function of $\sum_{i=1}^n A_i B_i C_i$.

Answer 1

Let $Z_i=A_iB_iC_i$, I first need to find the probability mass function of $Z_i$:

$Z_i$ is $0$ with probability $a$ regardless of the value of $B_i$ and $C_i$.

For $Z_i$ to be $1$, we need $A_i=1$ (which ocurrs with probability $a$) and the other two variables must be both $1$ or both $-1$. Thus, the probability of $Z_i=1$ is $a(1-b-c+2bc)$.

Finally, following a similar argument, the probability of $Z_i=-1$ is $a(b+c-2bc)$.

Thus, the moment generating function must be:

$$M_{Z_i}(t)=(1-a)+e^t a(1-b-c+2bc)+e^{-t}a(b+c-2bc)$$

Then, we obtain the moment generating function of the sum by $\prod M_{Z_i}(t)$.

Sanity check:

$E[Z_i] = M_{Z_i}'(0) = a(1-2(b+c-2bc))$

Empirically: