Calculate probabilities and expected value $\mathbb E(X)$?

Question

Let $(Ω, \mathbb P)$ be a probability space, given two independent random variables $X, Y$, which can take the value $0$ or $1$.

Let $\mathbb P(X=0) = p,\mathbb P(Y=0) = q$, with $p, q ∈ [0,1]$.
Additionally, we have the random variable $Z = 2(X+Y) \mod 3$

Calculate all the single probabilities of $Z$ and the expected value $\mathbb E(Z)$.

Answer 1

Just apply definitions and you are done. Recognize that $2(0+0)\equiv 0\pmod{3}$, that $2(0+1)\equiv 2\pmod{3}$, that $2(1+0)\equiv 2\pmod{3}$ and that $2(1+1)\equiv 1\pmod{3}$. Continue using the relationship between $Z$ and the joint distribution for $X,Y$, and apply what you know about independent random variables.

$Pr(Z=0) = Pr(X=0,Y=0)=Pr(X=0)\cdot Pr(Y=0) = pq$

$Pr(Z=1) = Pr(X=1,Y=1)=Pr(X=1)\cdot Pr(Y=1)=\dots$

$Pr(Z=2) = \dots$

$E[Z] = 0\cdot Pr(Z=0) + 1\cdot Pr(Z=1)+2\cdot Pr(Z=2) = 0\cdot pq + 1\cdot (\dots)+\dots$

Fill in the missing information.