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)012xP(X= 1 and Y 1) = [0(a) What is P(X= 1 and Y 1) ?Py(y)P(X ≤ 1 and Y≤1):y(b) Compute P(X≤ 1 and Y≤ 1).10.10 0.03 0.020.06 0.20 0.080.05 0.14 0.320(c) Give a word description of the event { X #0 and Y #0}.O One hose is in use on both islands.O One hose is in use on one island.O At most one hose is in use at both islands.O At least one hose is in use at both islands.Compute the probability of this event.P(X0 and Y / 0) =(d) Compute the marginal pmf of X.0Compute the marginal pmf of Y.21Using p(x), what is P(X≤ 1)?P(X ≤1)=[22(e) Are X and Y independent rv's? Explain.OX and Y are independent because P(x, y) ‡ px(x) · Py (y).OX and Y are not independent because P(x, y) #Px(1) - Py(Y).OX and Y are independent because P(x, y) = Px(2).py (y).OX and Y are not independent because P(x, y) = Px(z) Py(y).

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Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 28EQ

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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. y p(x, y) 0 1 2 x 0 0.10 0.03 0.02 1 0.07 0.20 0.08 2 0.05 0.14 0.31 (a) Given that X = 1, determine the conditional pmf of Y—i.e., pY|X(0|1), pY|X(1|1), pY|X(2|1). (Round your answers to four decimal places.) y 0 1 2 pY|X(y|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.) y 0 1 2 pY|X(y|2) (d) Given that two hoses are in use at the…
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