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 horses. Let X denote the number of horses being used on the self-service island at a particular time and let Y denote the number of horses on the full-service island in use at that time. The joint pmf of X and Y, p(x,y) is given below: y P(x,y) 0 1 2 x 0 .10 .04 .02 1 .08 .20 .06 2 .06 .14 .30 a) What is the probability that exactly one pump is used in each island? b) What is probability that at most one pump is used in self-service island and no pumps used in full-service island? c) Compute marginal pmfs of X and Y (pX(x) and pY(y)). d) What is the probability that at least one pump is used in full-service island?

Explanation / Answer

from above below is joint probability distirbution:

a)

probability that exactly one pump is used in each island =P(X=1;Y=1) =0.2

b)

probability that at most one pump is used in self-service island and no pumps used in full-service island

=P(X<=1 ;Y=0)=P(X=0;Y=0)+P(X=1;Y=0)=0.1+0.08 =0.18

c)

marginal pmfs of X :

marginal pmf of Y:

d)  probability that at least one pump is used in full-service island =P(Y=1)+P(Y=2) =0.38+0.38 =0.76

(Note: please reply if table looks differnt as given on top ; as values are scattered in problem)

y x 0 1 2 total 0 0.1 0.04 0.02 0.16 1 0.08 0.2 0.06 0.34 2 0.06 0.14 0.3 0.5 total 0.24 0.38 0.38 1

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