Two random variables have joint PMF pXYxy Find an expression

Two random variables have joint PMF pX,Y[x,y] Find an expression for Var(X + Y) with and without independence. Find an expression for Var(X - Y) with and without independence.

Solution

a)

For 2 independent variables, the formula is

Var(X+Y) = Var(X) + Var(Y)

For 2 dependent variables, the formula is

Var(X+Y)= Var(X) + Var(Y) + 2*Cov(X,Y)

n)

For 2 independent variables, the formula is

Var(X-Y)= Var(X)    +    Var(Y)

For 2 dependent variables, the formula is

Var(X - Y)   =   Var(X)   + Var(Y) - 2*Cov(X,Y)