Question

1. [45 points] Let the pair (X, Y) have joint PMF given by the following table XY | 0 | 1 | 2 --- | --- | --- | --- 0 | 1/9 | 2/9 | 1/9 1 | 2/9 | 2/9 | 0 2 | 1/9 | 0 | 0 (a) [6 points] Find the marginal PMFs. (b) [6 points] Find the probability of the event A = {X ? 1}, B = {X ? Y} and C = {X + Y = 2}. (c) [10 points] Find the conditional PMF of Y given X = x for x = 0, 1, 2. (d) [8 points] Find the conditional expectation of Y given X = x for x = 0, 1, 2, and verify the identity E[Y] = ?x fX(x)E[Y|X = x]. (e) [9 points] Compute E[XY]. (f) [6 points] Are X and Y independent? Why?

          1. [45 points] Let the pair (X, Y) have joint PMF given by the following table

XY | 0 | 1 | 2
--- | --- | --- | ---
0 | 1/9 | 2/9 | 1/9
1 | 2/9 | 2/9 | 0
2 | 1/9 | 0 | 0

(a) [6 points] Find the marginal PMFs.
(b) [6 points] Find the probability of the event A = {X ? 1}, B = {X ? Y} and C = {X + Y = 2}.
(c) [10 points] Find the conditional PMF of Y given X = x for x = 0, 1, 2.
(d) [8 points] Find the conditional expectation of Y given X = x for x = 0, 1, 2, and verify the identity
E[Y] = ?x fX(x)E[Y|X = x].
(e) [9 points] Compute E[XY].
(f) [6 points] Are X and Y independent? Why?

1. [45 points] Let the pair (X, Y) have joint PMF given by the following table

XY | 0 | 1 | 2
— | — | — | —
0 | 1/9 | 2/9 | 1/9
1 | 2/9 | 2/9 | 0
2 | 1/9 | 0 | 0

(a) [6 points] Find the marginal PMFs.
(b) [6 points] Find the probability of the event A = X ? 1, B = X ? Y and C = X + Y = 2.
(c) [10 points] Find the conditional PMF of Y given X = x for x = 0, 1, 2.
(d) [8 points] Find the conditional expectation of Y given X = x for x = 0, 1, 2, and verify the identity
E[Y] = ?x fX(x)E[Y|X = x].
(e) [9 points] Compute E[XY].
(f) [6 points] Are X and Y independent? Why?

Added by Sarah C.

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