8. A service station has both self-service and full-service islands. On each isl

Question

8. 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 used on the self-service island at a particular time, and let Y denote the num on t ber of hoses he full-service island in use at that time. The joint probability mass function of X and Y is given below. Px) o 1 2 0 0.10 0.04 0.02 x 0.08 0.20 0.06 2 0.06 0.14 0.30 (a) Find P(XSI, Y-2) Answer o O% (b) Find the marginal distribution of X. pta lo16 lo-y lo.so (c) Find the expected value of Y. AnswerIu

Explanation / Answer

a) The required probability here is computed as:

P(X <= 1, Y = 2) = P(X = 0, Y = 2) + P(X =1, Y = 2) = 0.02 + 0.06 = 0.08

Therefore 0.08 is the required probability here.

b) The marginal PDF for X here is obtained by adding the corresponding rows as:

P(X = 0) = 0.16,
P(X = 1) = 0.34,
P(X = 2) = 0.50

c) The expected value of Y here is obtained as:

E(Y) = 0 + 1*0.38 + 2*0.38 = 1.14

Therefore 1.14 is the expected value of Y here.

d) The second moment of Y here is obtained as:

E(Y2) = 0 + 12*0.38 + 22*0.38 = 1.9

Now the variance is computed as:

Var(Y) = E(Y2) - [ E(Y)]2 = 1.9 - 1.142 = 0.6004

Therefore 0.6004 is the required variance here.

e) The expected value of XY here is computed as:

E(XY) = 1*0.2 + 2*(0.06 + 0.14) + 4*0.3 = 1.8

Therefore 1.8 is the expected value here.

f) P(X = 0) = 0.16, P(Y = 0) = 0.24

P(X = 0)P(Y = 0) = 0.16*0.24 = 0.0384

P(X = 0, Y = 0) = 0.10 which is not equal to P(X = 0)P(Y = 0)

Therefore X and Y are not independent here.

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