Binomial distribution Question about the product of two individual random variables How do you solve this question:
Consider two independent binomially-distributed random variables $X \sim B(2, a)$ and $Y \sim B(2, b)$. Let $W$ be the random variable that represents the product of each value of $X$ with each value of $Y$. Construct a table showing the probability distribution of $W$. Hence find an expression for $\mathrm{E}(W)$ in terms of $a$ and $b$.

$X$ and $Y$ take on values 0,1,2, so $XY$ takes on values $0,1,2,3,4$.
The probability of $P(W=w) = \sum_{x,y: xy=w} P(X=x,Y=y) = \sum_{x,y: xy = w} P(X=x) P(Y=y)$.
For example, $P(W=0)$ will involve the $(x,y)$ pairs: $(0,0),(0,1),(0,2),(1,0),(2,0)$.
Then, $E[W] = \sum_{w=0}^4 P(W=w)$. 
