Find the value of $E[X^2Y]$.

Question

Let $X\in\{0,1\},Y\in\{3,4\}$ be discrete random variables, and their joint distrubtion be given as in the following table:

$$\begin{array}{c|c|c|} X/Y& \text{3} & \text{4} \\ \hline \text{0} & \frac{1}{4} & \frac{3}{8} \\ \hline \text{1} & \frac{1}{4} & \frac{1}{8} \\ \hline \end{array}$$

Find $E[X^2Y]$.

What I did:

We have that $\sum x_i^2y_iP(X^2Y=x_i^2y_i)=\sum x_i^2y_i P(X^2=x_i^2,Y=y_i)$

What do I do with $X^2$? Does it take the same probabilities as $X$ when calculating the $P(X^2=x_i,Y=y_i)?$

Answer 1

Observe that $$ E(X^2Y)=\sum x^2yP(X=x,Y=y) $$ where the summation is over the set $(x,y)\in\{0,1\}\times\{3,4\}$ and $P(X=x,Y=y)$ can be found from your joint distribution table.