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

Question

Mathematics

Jimena

29 January, 22:44

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

+4

Answers (1)

Know the Answer?

Sofia Dalton · Answer 1

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