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. px, y) 0 2 0 0.10 0.05 0.02 1 0.06 0.20 0.08 2 0.06 0.140.29 (a) Given thatX = 1 , determine the conditional pmf of Y_i.e., Prix(011), Prix(111), pYlx(211). (Round your answers to four decimal places.) 2 Pyx(1) (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.) 2 Pyxrl) (c) Use the result of part (b) to calculate the conditional probability P(Y S 1 X = 2). (Round your answer to four decimal places.)

Explanation / Answer

a) From the table, we get: P(X = 1) = 0.06 + 0.2 + 0.08 = 0.34

Therefore the conditional distribution for Y given X is computed as:

b) Now given X = 2, the conditional probability of Y is computed in the same way.

We have here: P(X =2 ) = 0.06 + 0.14 + 0.29 = 0.49

Therefore the conditional distribution of Y is computed as:

c) Now from the above table, the required probabiltiy is computed as:

P( Y < =1 | X = 2) = P(Y = 0 | X=2) + P(Y = 1 | X=2) = 0.1224 + 0.2857 = 0.4081

Therefore 0.4081 is the required probability here.

Y=y 0 1 2 P(Y=y | X = 1) 0.06/0.34 = 0.1765 0.2/0.34 = 0.5882 0.08/0.34 = 0.2353

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