1 A service station has both selfservice and full service is

1. A service station has both self-service and full service islands. On each island there are two pumps. Let X be the number of pumps used on the self-service island, and Y be the number of pumps used on the full service island. The following is the joint probability mass function of X and Y.

X\\Y

0

1

2

0

0.11

0.03

0.02

1

0.09

0.19

0.06

2

0.06

0.14

0.30

(a) Evaluate P(X>1, Y<1)

(b) Evaluate P(X>1.5, Y<1.5)

(c) Evaluate P(X>1)

(d) Calculate the marginal probabilities for X and Y.

(e) Are X and Y independent? Explain.

(f) Calculate the correlation between X and Y.

Solution

1) P(X>1, Y<1 ) = P (X = 2, Y =0) = 0.06

2) P(X>1.5, Y<1.5) = P (X = 2, Y =0) = 0.06

3) P(X>1) = P(X = 2) = P(X = 2, Y =0) +P(X = 2, Y =1) + P(X = 2, Y =2) = 0.5

4) Marginal Probabilities for X .

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

on the same lines by taking summation over Y - values we get

P(X = 1 ) = 0.34 and P(X =2) = 0.5

Marginal Probabilities for Y .

P(Y = 0 ) =  P(X = 0, Y =0) +P(X =1, Y =0) + P(X =2, Y =0) = 0.26

on the same lines by taking summation over X- values we get

P(Y = 1 ) = 0.36 and P(Y =2) = 0.38

5) IS X and Y independent .

Two events X and Y are said independent if

P(X = x, Y =y) = P(X = x) * P(Y =y)

L.H.S = P( X= 0, Y = 0) = 0.11 and R.H. S = P(X = 0 ) * P(Y = 0) = 0.0416

As, L.H.sS is not equal to R.H.S

X and Y are not independent.

6) corr (X, Y) = E(XY) - E(X) E(Y) / Std (X) * Std(Y)

E(XY) = 1.79, E(X) = 1.34 , E(Y) = 1.12, Std (Y) = 1.3711 , Std (X) = 1.529

corr (X, Y) = 0.138