Combining independent distributions

Question

Suppose that $X_{1} \sim B(n_{1},p_{1})$ and $X_{2} \sim B(n_{2},p_{2})$ are independent. Write down the moment generating function for the random variable $Y=X_{1}+X_{2}$.

I know the moment generating function $M_{Y}(t)$ is defined at $E[e^{tY}]$.

$$E[e^{tY}] = \sum^{\infty}_{y=0}e^{ty}f_{Y}(y)$$ $$= \sum^{n_{1}+n_{2}}_{x_{1}+x_{2}=0}e^{t(x_{1}+x_{2})}f_{X_{1}+X_{2}}(x_{1}+x_{2})$$

But now I'm confused in what to do next.

Answer 1

When you add independent RV's $X$ and $Y$, your moment generating functions multiply: $E[e^{t (X+Y)}] = E[e^{tX}e^{tY}] = E[e^{tX}]E[e^{tY}]$ where the last step follows by independence of $X$ and $Y$ implies functions of them are independent (so $e^{tX}$ and $e^{tY}$ are independent).

Answer 2

Hint: If $X_1$ and $X_2$ are independent: $$ \mathbb E(e^{tX_1}e^{tX_2})=\mathbb E(e^{tX_1})\mathbb E(e^{tX_2}). $$