Let $X$, $Y$, and $Z$ be independent random variables, where X is Bernoulli with parameter $1/3$, $Y$ is exponential with parameter $2$, and $Z$ is Poisson with parameter $3$.
(a) Consider the new random variable $U = XY + (1 - X)Z$ . Find the transform associated with $U$.
(b) Find the transform associated with $2Z + 3$.
(c) Find the transform associated with $Y + Z$.
I know that I can get the transform of a sum of random variables by multiplying their transforms but I don't understand how I can get the complete transform of $U$.
If $X=0$, the rv U is discrete and it is a poisson $Po(3)$. This happens with probabilty $\frac{2}{3}$
If $X=1$, the rv U is continuous and it is a $Exp(2)$. This happens with probabilty $\frac{1}{3}$
Thus
$$f_U(u)=\frac{2}{3}e^{-2u}\cdot\mathbb{1}_{[0;+\infty)}(u)+\frac{2}{3}\frac{e^{-3}3^u}{u!}\cdot\mathbb{1}_{\{0;1;2;3;...\}}(u)$$
Show your attempts for the rest of the questions (if you want that someone helps you)