A service station has both self-service and full-service islands. On each island

Question

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 2 0 0.10 0.03 0.02 10.08 0.20 0.08 2 0.05 0.14 0.30 (a) Given that X = 1, determine the conditional pmf of y-i.e., pYlx(011), pnx(111), pYlx(211). (Round your answers to four decimal places.) 2 Prix(yl1) (b) Given that two hoses are in use at the self-service island, what is the conditional pmf of the number of hoses in use on the full-service island? (Round your answers to four decimal places.) Pyx(12) (c) Use the result of part (b) to calculate the conditional probability P(Y 1 X-2). (Round your answer to four decimal places.) (d) Given that two hoses are in use at the full-service island, what is the conditional pmf of the number in use at the self-service island? (Round your answers to four decimal places.) 0 2 Pxi x12) Need Help? LReadit 1Talkto aTutor1

Explanation / Answer

a) The conditional probabilities here are computed as:

P(X = 1) = 0.08 + 0.2 + 0.08 = 0.36

Therefore, we get the conditional distribution as:

b) Similarly, here we have:

P(X = 2) = 0.05 + 0.14 + 0.3 = 0.49

Therefore, the conditional distribution here would be given as:

c) From the above distribution, we get:

P( Y < = 1 | X=2) = P(Y = 0 | X=2) + P( Y =1 | X = 2) =0.1020 + 0.2857 = 0.3877

Therefore 0.3877 is the required probability here.

d) Now here using the same method, first we compute:

P( Y = 2) = 0.02 + 0.08 + 0.3 = 0.4

Y 0 1 2 P(Y | X=1) 0.08/0.36 = 0.2222 0.20/0.36 = 0.5556 0.08/0.36 = 0.2222

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