A certain market has both an express checkout line and a super-express checkout

Question

A certain market has both an express checkout line and a super-express checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X1 and X2 is as given in the accompanying table.

The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout is X1 X2. Calculate the expected difference

x2   0     1     2     3   x1   0   0.08 0.06 0.04 0.00   1   0.07 0.14 0.05 0.05   2   0.05 0.04 0.10 0.06   3   0.00 0.03 0.04 0.07   4   0.00 0.01 0.05 0.06

Explanation / Answer

From the given Joint PMF, we will calculate Marginal PDF of both the variable X1 and X2

fX1(X1 ) = 0.18 ; x1 = 0

= 0.31 ; x1 = 1

= 0.25; x1 = 2

= 0.14; x1 = 3

= 0.12; x1 = 4

E(X1) = 0 * 0.18 + 1 * 0.31 + 2 * 0.25 +3 * 0.14 + 4 * 0.12 = 1.71

fX1(X1 ) = 0.20 ; x2 = 0

= 0.28 ; x2 = 1

= 0.28; x2 = 2

= 0.24; x = 3

E(X2) = 0 * 0.20 + 1 * 0.28 + 2 * 0.28 +3 * 0.24 = 1.56

so E(X1 - X2) = 1.71 -1.56 = 0.15

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