Let X and Y two independent variables, U = XY and V = X + Y

Question

Let $X,Y$ be two independent variables distributed with Bernoulli distribution with coefficient 1/3. The Bernoulli distribution is the following: X = 1 with probability 1/3 and X = 0 with probability 2/3. Can I say something about the covariance of U and V? Let E[X] denote the expected value, then I know that $$ \operatorname{Cov}[U,V] = E[UV] - E[U]*E[V] $$ But then I do not know how to calculate E[UV]. Any suggestions?

Answer 1

$E(UV) = E(X^2Y+Y^2X) = E(X^2Y)+E(Y^2X)$.

What do we know about the distribution of $X^2Y$. $X^2Y=1$ when both $X=1$ and $Y=1$, and is 0 otherwise. $X^2Y=1$ with probability $1/9$ (This is a bernoulli). Same goes for $Y^2X$.

Answer 2

You can do it symbolically, as in pbn990's answer. Alternatively, you can draw up a table:

$$ \begin{array}{c|c|c|c||c|c} X & Y & U & V & UV & \text{probability} \\ \hline 0 & 0 & 0 & 0 & 0 & 4/9 \\ \hline 0 & 1 & 0 & 1 & 0 & 2/9 \\ \hline 1 & 0 & 0 & 1 & 0 & 2/9 \\ \hline 1 & 1 & 1 & 2 & 2 & 1/9 \\ \end{array} $$

Since $UV = 2$ with probability $1/9$ and is equal to $0$ otherwise, $E(UV) = 2 \times 1/9 = 2/9$.