5. +-112 points DevoreStat9 5.E.001 My Notes Ask Your Teacher A service station

Question

5. +-112 points DevoreStat9 5.E.001 My Notes Ask Your Teacher A service station has both self-service and full-service islands. On each island, there is a single regular unleaded pump with two hoses. Let X denote the number of hoses being used on the self-service island at a particular time, and let Y denote the number of hoses on the full-service island in use at that time. The joint pmf of X and Y appears in the accompanying tabulation p(x, y) 0 0.10 0.03 0.02 1 0.08 0.20 0.06 2 0.06 0.14 0.31 (a) What is P(X 1 and Y-1)? P(X = 1 and Y= 1)- (b) Compute P(X S 1 and Y S 1) P(X 1 and Y 1) = (c) Give a word description of the event {X # 0 and Y # 0). At most one hose is in use at both islands. At least one hose is in use at both islands. One hose is in use on both islands

Explanation / Answer

a) P(X = 1 and Y = 1) = 0.2

b) P(X < 1 and Y < 1) = P(X = 0 and Y = 0) + P(X = 0 and Y = 1) + P(X = 1 and Y = 0) + P(X = 1 and Y = 1)

                                   = 0.1 + 0.03 + 0.08 + 0.2

                                   = 0.41

c) Option-B) at least one house is in use at both islands

P(X 0 and Y 0) = 1 - (P(X = 0 and Y = 0) + P(X = 0 and Y = 1) + P(X = 1 and Y = 0) + P(X = 0 and Y = 2) + P(X = 2 and Y = 0))

                               = 1 - (0.1 + 0.03 + 0.08 + 0.02 + 0.06)

                               = 0.71

d) P(X = 0) = 0.1 + 0.03 + 0.02 = 0.15

P(X = 1) = 0.08 + 0.2 + 0.06 = 0.34

P(X = 2) = 0.06 + 0.14 + 0.31 = 0.51

P(Y = 0) = 0.1 + 0.08 + 0.06 = 0.24

P(Y = 1) = 0.03 + 0.2 + 0.14 = 0.37

P(Y = 2) = 0.02 + 0.06 + 0.31 = 0.39

P(X < 1) = P(X = 0) + P(X = 1) = 0.14 + 0.35 = 0.49

e) Option-A)

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