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 0 0.07 0.04 0.09 0.04 0.11 0.2 0 0.12 a.) Find a. (Give decimal answer to two places past the decimal.) 0.2300 You are correct Your receipt no. is 168-297 b.) Compute P(X 1 and Y 2). (Give decimal answer to two places past the decimal.) 0.1100 You are correct Your receipt no. is 168-9474 c.) What is P(XS1)? (Give decimal answer to two places past the decimal.) 0.5900 You are correct Your receipt no. is 168-18356 d.) what is P(X=1|Y=2)? (Give decimal answer to two places past the decimal.) 0.3100 You are correct Your receipt no. is 168-6859 e.) what is E(Y|X=0)? (Give decimal answer to two places past the decimal.) 1.15 Submit AnswerIncorrect. Tries 3/5 Previous Tries f.) What is Var(YX-0)? (Give decimal answer to two places past the decimal.) Submit Answe Tries 0/5 g.) Are X and Y independent random variables? Yes No

Explanation / Answer

P(X = 0) = 0.07 + 0.1 + 0.04 = 0.21

Given X = 0 , The PDF of the distribution here would be obtained as:

P(Y = 0 | X= 0) = 0.07 / 0.21 = 0.3333
P(Y = 1 | X = 0) = 0.1 / 0.21 = 0.4762
P( Y = 2 | X = 0) = 0.04 / 0.21 = 0.1905

e) The expected value here is computed as:

E(Y | X = 0) = 0*P(Y = 0 | X= 0) + 1*P(Y = 1 | X = 0) + 2*P( Y = 2 | X = 0) = 0.4762 + 2*0.1905 = 0.8572

Therefore E(Y | X = 0) = 0.86

f) The second moment here is computed as:

E(Y2 | X = 0) = 0.4762 + 22*0.1905 = 1.2382

Therefore, we get the conditional variance as:

Var(Y | X =0) = E(Y2 | X = 0) - [ E(Y | X = 0) ]2 = 1.2382 - 0.85722 = 0.50

Therefore we get: V(Y | X=0) = 0.50

g) Here we have:

P(X = 0) = 0.21 and P(Y = 0) = 0.07 + 0.04 + 0.09 = 0.2

P(X = 0)P(Y = 0) = 0.21*0.2 = 0.042 which is not equal to P( X = 0 , Y =0)

Therefore the 2 variables are not independent here.

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