Poisson random variable of two independent variables

Question

I have been given two independent Poisson random variables $X$ and $Y$, with parameters $a$ and $b$. There is random Poisson variable $Z$ such that:

1. $Z = X + Y$
2. $Z = X + 2$
3. $Z = 3X$
4. $Z = XY$

I have to show which ones are true, and if they are, find the parameter for $Z$.

I did the first one using the summation property and it is true, so parameter $= a+b$. But for others I am not sure. Second one looks false (gut feeling), and third one true. No idea about the 4th one. Any help will be much appreciated.

TIA!

Answer 1

• For the second, ask yourself the question: what is the probability that $Z=0$? (And remember that a Poisson r.v. with parameter $\lambda> 0$ has probability $e^{-\lambda}>0$ to be equal to $0$).

• For the third, ask yourself: what is the probability that $Z=2$? (And remember that a Poisson r.v. with parameter $\lambda> 0$ has probability $\frac{\lambda^2}{2} e^{-\lambda}>0$ to be equal to $2$).

• For the fourth, ask yourself two things: (1) what is the probability that $Z=1$? (And remember that a Poisson r.v. with parameter $\lambda> 0$ has probability $\lambda e^{-\lambda}$ to be equal to $1$). And (2) what is $\mathbb{E}[Z]$ (using independence of $X,Y$)? Note that if $Z$ is Poisson, you know exactly its parameter (as a function of $a,b$) since it is its expectation. Now check if those two things are compatible.