given any credit. Aservice station has both self-service and ful-service islands

Question

given any credit. Aservice station has both self-service and ful-service islands on each island, there is ssingle egular unleaded pump with two hoses. Let Xdenote e number hoses being used on the self-service island at a particular time, and letYdenote the number of hoses cn the ful-service island in use at a particular time. The joint probability distribution function of X and Y is 1 08 20 06 06 14 30 a. what is POxsi y 1) p (0,0 b Compute the marginal distribution functions for x and r Arexand Y ustify your

Explanation / Answer

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

(b) P(X=0) = P(X=0, Y=0) + P(X=0, Y=1) + P(X=0, Y=2) = 0.16

P(X=1) = P(X=1, Y=0) + P(X=1, Y=1) + P(X=1, Y=2) = 0.34

P(X=2) = P(X=2, Y=0) + P(X=2, Y=1) + P(X=2, Y=2) = 0.50

Therefore the marginal distribution of X is

P(X less than or equal to x)= 0.16, x less than or equal to 0

= 0.50, x less than or equal to 1

= 1, x less than or equal to 2

P(Y=0) = P(X=0, Y=0) + P(X=1, Y=0) + P(X=2, Y=0) = 0.24

P(Y=1) = P(X=0, Y=1) + P(X=1, Y=1) + P(X=2, Y=1) = 0.38

P(Y=2) = P(X=0, Y=2) + P(X=1, Y=2) + P(X=2, Y=2) = 0.38

Therefore the marginal distribution of Y is

P(Y less than or equal to y)= 0.24, y less than or equal to 0

= 0.62, y less than or equal to 1

= 1,   y less than or equal to 2

NO, X and Y are not independent.

P(X=0,Y=0)=0.1, P(X=0)*P(Y=0)=0.16*0.24

P(X=0,Y=0) not equal to P(X=0)*P(Y=0). Hence, X and Y are not independent.

(c) Conditional density of X given Y=2 = P(X, Y=2)/P(Y=2) = P(X, Y=2)/0.38

Conditional density of X=0 given Y=2 = P(X=0, Y=2)/0.38 =0.02/0.38 = 1/19

Conditional density of X=1 given Y=2 = P(X=1, Y=2)/0.38 =0.06/0.38 = 3/19

Conditional density of X=2 given Y=2 = P(X=2, Y=2)/0.38 =0.30/0.38 = 15/19

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