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 $b$.

Hint: 
$P(X=k_1,Y=k_2)=P(X=k_2)\cdot P(Y=k_2)=f_{X,Y}(x,y)$
$=\binom{n_1}{k_1}\binom{n_2}{k_2}\cdot a^{k_1}\cdot (1-q)^{n_1-k_1}\cdot b^{k_2}\cdot (1-b)^{n_2-k_2}$
with $X \sim B(n_1,a)$ and $Y \sim B(n_2,b)$
For $n_1=n_2=2$ you need a 3x3 table.
The expected value is $E(XY)=\sum_{k_1=0}^2  \sum_{k_2=0}^2 k_1 \cdot k_2\cdot f_{X,Y}(x,y)$ 
$=\sum_{k_1=0}^2  \sum_{k_2=0}^2 k_1 \cdot k_2\cdot \binom{2}{k_1}\binom{2}{k_2} a^{k_1}\cdot (1-a)^{2-k_1}\cdot b^{k_1}\cdot (1-b)^{2-k_2}$
