Q:

Let Y1 = X1 + X2 and Y2 = X1 - X2. What's the covariance between Y1 and Y2?

Given:
$$F_X(\vec x) = (x_1+x_2)/9 \quad \Big[x_1 \in \{-1,0,1\}, x_2 \in\{ 1,2\}\Big]$$

Mean of $X_1 = 4/9$
Variance of $X_1= 38/81$

Given: Mean of $X_2 = 5/3$
Variance of $X_2= 2/9$

I solved for: $E[Y_1] = 19/9$ and $E[Y_2] = -11/9$
What is the next step to find covariance? I know that finding covariance between two $X$ values is $E[X_1X_2] - μ_1μ_2$, but don't know how to apply that formula to $E[Y_1Y_2] - μ_{y_1}μ_{y_2}$.
Could someone please explain the difference between expectation of $Y_1Y_2$ and mean of $Y_1$ and $Y_2$ and provide steps on how to solve this problem? The answer is $20/81$, but I am looking for the process of getting to that answer.

A:

Hint: $Cov(Y_1,Y_2) = Cov(X_1+X_2,X_1-X_2) = Cov(X_1,X_1)-Cov(X_1,X_2)+Cov(X_2,X_1) - Cov(X_2,X_2)$
But what is the covariance of a random variable with itself?

