I am currently dealing with the following exercise:
Given the random variables $X \sim Be(p), Y \sim Exp(\lambda)$, and assume they are independent. Set $Z:= X + Y$.

• Compute the Moment Generating Function (MGF) $m_{z}(t)$ of $Z$
• Compute the Distribution Function (CDF) $F_{z}$ of $Z$. Is $Z$ an absolutely continuous random variable?

From the theory I have studied, I have some questions regarding how to deal with this kind of exercises:

1. When it comes to sum two discrete independent random variables and then calculating their CDF, does this always mean using convolution? Does this differ from calculating their joint PDF?
2. If X depended on Y, the joint CDF would be obtainable by using the law of total probability?
3. Since they are independent, the joint MGF is simply the product of the two MGFs?

I know it may sound odd, but I still have hard times finding a way to approach to these exercises. Thanks.