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.

p(x, y)

(a) What is P(X = 1 and Y = 1)?

(b) Compute P(X 1 and Y 1).

(c) Give a word description of the event {X 0 and Y 0}.

Compute the probability of this event.

(d) Compute the marginal pmf of X.

Compute the marginal pmf of Y.

Using pX(x), what is P(X 1)?

(e) Are X and Y independent rv's? Explain.

y

p(x, y)

     0 1 2 x 0     0.10     0.03     0.01   1     0.07     0.20     0.07   2     0.06     0.14     0.32  

Explanation / Answer

a) Here from the table, we get:

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

Therefore 0.2 is the required probability here.

b) The required probability here is computed as:

P( X < = 1 and Y < =1 ) = P(X = Y = 0 ) + P(X=1, Y =0) + P(X=0, Y =1) + P(X = Y = 1) = 0.1 + 0.03 + 0.07 + 0.2 = 0.4

Therefore 0.4 is the required probability here.

c) P(X not equal to 0 and Y not equal to 0 ) is described as: the number of hoses being used on the self-service island at a particular time is not equal to 0, and the number of hoses on the full-service island in use at that time is also not equal to 0

d) The marginal PMF for X can be computed by summing up the rows as:

P(X = 0 ) = 0.14, P(X =1) = 0.34 and P(X = 2) = 0.52

The marginal PMF for Y can be computed by summing up the columns as:

P(Y = 0 ) = 0.23, P(Y =1) = 0.37 and P(Y = 2) = 0.40

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