Finding covariance using $E[X]$ and $E[Y]$

Question

I have a table

where I have found $E[X] = 2.6$ and $E[Y] = 2.44$.

\begin{align*} E[X] &= 2 \times 0.4 + 3 \times 0.6 = 2.6 \\ E[Y] &= 2 \times 0.56 + 3 \times 0.44 = 2.44 \end{align*}

I have tried:

$$ Cov[X,Y] = E[XY] - E[X] E[Y]$$

How do I solve for covariance between $X$ and $Y$?

Answer 1

Using the formula you shared

$$Cov[X,Y] = E[X*Y] - E[X] * E[Y]$$

you have computed both $E(X),E(Y)$ . Now to compute $E(XY)$ treat $XY$ as a single random variable, ie the outcomes are $2\times 3=6$, $2\times 2=4 $ and $3\times 3 =9$. Get their respective probabilies from the table and compute the average as you did.