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 ? appears in the accompanying tabulation p(x, y) 0 0 0.10 0.05 0.02 1 0.08 0.20 0.07 2 0.05 0.14 0.29 (a) Given that X = 1, determine the conditional pmf of y-i.?., pYlx(011), pYly(11), pYlx(211). (Round your answers to four decimal places.) 0 Pyxil1) (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.) 0 Pyxy12) (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

Explanation / Answer

P(X=1)=P(X=1,Y=0)+P(X=1,Y=1)+P(X=1,Y=2)=0.08+0.2+0.07=0.35

now

P(Y=k|X=1)=P(Y=k,X=1)/P(X=1)

hence

P(Y=0|X=1)=P(Y=0,X=1)/P(X=1)=0.08/0.35=0.2286

P(Y=1|X=1)=P(Y=1,X=1)/P(X=1)=0.2/0.35=0.5714

P(Y=2|X=1)=P(Y=2,X=1)/P(X=1)=0.07/0.35=0.2

(b)

P(X=2)=P(X=2,Y=0)+P(X=2,Y=1)+P(X=2,Y=2)=0.05+0.14+0.29=0.48

now

P(Y=k|X=2)=P(Y=k,X=2)/P(X=2)

so

P(Y=0|X=2)=P(Y=0,X=2)/P(X=2)=0.1042

P(Y=1|X=2)=P(Y=1,X=2)/P(X=2)=0.2917

P(Y=2|X=2)=P(Y=2,X=2)/P(X=2)=0.6041

(c)

P(Y<=1|X=2)=P(Y=0|X=2)+P(Y=1|X=2)=0.3959

(d)

P(Y=2)=P(X=0,Y=2)+P(X=1,Y=2)+(X=2,Y=2)=0.02+0.07+0.29=0.38

now

P(X=k|Y=2)=P(X=k,Y=2)/P(Y=2)

so

P(X=0|Y=2)=P(X=0,Y=2)/P(Y=2)=0.0526

P(X=1|Y=2)=P(X=1,Y=2)/P(Y=2)=0.07/0.38=0.1842

P(X=2|Y=2)=P(X=2,Y=2)/P(Y=2)=0.29/0.38=0.7631

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