Here are the PMFs for three different discrete random variables, X, Y, and Z. PMF of X PMF of Y PMF of Z 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -1 1 2 3 4 5 0 X 0 0 z (a) For each part below, circle the value that is the largest of the three and underline the value that is the smallest of the three. In part (iii), F refers to the CDF of each random variable. You do not need to justify your answers, as there is no partial credit for this question. Hint: you do not need to exactly compute all of these values in order to answer the question! (8 points) i. E[X] E[Y] E[Z] ii. var(X) var(Y) var(Z) iii. FX(-1) Fy(-1) Fz(-1) iv. P(X = 1 X > 0) P(Y = 1 Y > 0) P(Z = 112 > 0) (b) Let W = , where the PMF of Z is given above. Compute E[W]. (2 points) Here are the PMFs for three different discrete random variables, X, Y, and Z. PMF of X PMF of Y PMF of Z 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -1 1 2 3 4 5 0 X 0 0 z (a) For each part below, circle the value that is the largest of the three and underline the value that is the smallest of the three. In part (iii), F refers to the CDF of each random variable. You do not need to justify your answers, as there is no partial credit for this question. Hint: you do not need to exactly compute all of these values in order to answer the question! (8 points) i. E[X] E[Y] E[Z] ii. var(X) var(Y) var(Z) iii. FX(-1) Fy(-1) Fz(-1) iv. P(X = 1 X > 0) P(Y = 1 Y > 0) P(Z = 112 > 0) (b) Let W = , where the PMF of Z is given above. Compute E[W]. (2 points)