The pair of random variables (X, Y) is equally likely to take any of the four pairs of values (0,1), (1,0), (−1,0), (0,−1). Note that X and Y each have zero mean.

a) Find E[XY].
E[XY]=
b) YES or NO: For this pair of random variables (X, Y), is it true that Var(X+Y)=Var(X)+Var(Y)?

Select an option Yes No
c) YES or NO: We know that if X and Y are independent, then Var(X+Y)=Var(X)+Var(Y). Is the converse true? That is, does the condition Var(X+Y)=Var(X)+Var(Y) imply independence?

Select an option Yes No

Answer and Step-by-step explanation:

(a)

E[XY] = 1/4*0*1 + 1/4*1*0 + 1/4*-1*0 +1/4*0*-1 = 0

(b)

E[X] = 0, E[Y] = 0

Thus, Cov(X,Y) = E[XY] - E[X]E[Y] = 0

So, Var(X + Y) = Var(X) + Var(Y) is True

The answer is Yes

(c) No, the converse is not true

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