Question

.Let X be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2; let Y be a random variable, independent of X, that takes the value 1 with probability 1/2, and takes the value −1 with probability 1/2; and let Z = XY . Show that Cov(X, Z) = 0. Show that X and Z are dependent

Answer 1

X = 1 with p =1/2

0   with p = 1/2

Y = 1 with p = 1/2

    -1 with p = 1/2

Z = XY

X/Y	1	-1
0	0.25	0.25
1	0.25	0.25

Z	p
0	0.5
1	0.25
-1	0.25

Cov(X,Z) = E(XZ) - E(X)E(Z)

joint distribution of X and Z

X/Z	0	1	-1
0	0.5	0	0
1	0	0.25	0.25

hence

Cov(X,Z) = 0

X/Z	0	1	-1
0	0.5	0	0	0.5
1	0	0.25	0.25	0.5
	0.5	0.25	0.25	1

P(X = 0, Z =0) = 0.5

P(X = 0) = 0.5 and P(Z = 0) = 0.5

P(X = 0) * P(Z =0) = 0.25

since P(X = 0, Z =0) P(X = 0) * P(Z =0)

X and Z are dependent

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