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 singlc regular unleaded pump with two hoses. Let X denote the number of hoses being used on the sclf 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. ptx, y) 0 0.10 0.04 0.02 10.06 0.20 0.07 2 0.06 0.14 0.31 (a) what is P(X = 1 and Y = 1)? P(X- 1 and Y 1) (b) Compute P(X 1 and Ys1) P(X s 1 and Ys 1) (c) G ve a word description of the event (X·0 and * o One hose is in use on one island. o One hose is in use on both islands At most one hose is in use at both islands. O At Icast one hose is in use at both islands. Compute the probability of this event. (d) Compute the marginal pmf of X. Px(x) Compute the marginal prnf of Y. PMy) Using pxtx), what is PX S 1)? (e) Are X and Y independent rv's? Explain O X and Y are not independent because P(x,y)s PX(x) , 1(y). @ and Yare independent because p(x.y)a PX(x) pny). O and Y are not independent because P(x,y)-PX(x)-pny). o Xand Y are independent because Px,y) Pxx) p

Explanation / Answer

a) From the given joint probability distribution tables, we get:

P(X = 1, Y = 1) = 0.2

b) P(X < =1, Y < =1 ) = P(X = 0, Y =0 ) + P(X=1, Y =0 ) + P(X=0, Y =1 ) + P(X = 1, Y = 1)

P(X < =1, Y < =1 ) = 0.1 + 0.04 + 0.06 + 0.2 = 0.4

Therefore 0.4 is the required probability here.

c) X not equal to 0 and Y not equal to 0 means that X > 0 and Y > 0 that is at least one hose in use at both islands.

The required probability here is computed as:

P( X not equal to 0 and Y not equal to 0 ) = 1 - P(X = Y = 0 ) = 1 - 0.1 = 0.9

Therefore 0.9 is the required probability here.

d) The marginal PMF for X here is computed as: ( by adding each of the individual rows )

Similarly, The marginal PMF for Y here is computed as: ( by adding each of the individual columns )

From the above tables, we get the required probability as:

P(X < =1 ) = P(X = 0) + P(X = 1) = 0.16 + 0.33 = 0.49

Therefore 0.49 is the required probability here.

e) From the above computations, we know that:

P( X = 1) = 0.33 and P(Y = 1) = 0.38 but we know that P(X = Y = 1) = 0.2

And also P(X =1)P(Y=1) = 0.33*0.38 = 0.1254 which is not equal to P(X = Y = 1) = 0.2

Therefore X and Y are not independent. Therefore a is the correct answer here.

X 0 1 2 p(X) 0.16 0.33 0.51

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