The number of withdrawals a bank processes in a day follows a random variable X. The number of deposits in a day is represented by $Y$. $X$ and $Y$ are independent and have the following moment generating functions
$$\displaystyle {M}_{X}(t)=\frac{p}{1-q{e}^{t}}$$ $$\displaystyle {M}_{Y}(t)={\left(\frac{p}{1-q{e}^{t}}\right)}^{r}$$ $q = 1 - p$
What is the expected number of transactions in a day?
Would I multiply these two together and then take the derivative and make all t's = $0$? I am a little shaky on Moment Generating Functions.
Hint:
$$\mathbb{E}[X+Y] = \mathbb{E}[X] + \mathbb{E}[Y]$$
You just have to compute $\mathbb{E}[X]$ and $\mathbb{E}[Y]$ separately.
Alternatively, since $X$ and $Y$ are independent, mgf of the independent sum is the product of mgf, you can also do what you suggested.