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 x_1 denote the number of customers in line at the express checkout at a particular time of cay, and let X_2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of and X_1 and X_2 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 X_1 - X_2. Calculate the expected difference.

Explanation / Answer

Number of customers in express checkout = X1

Number of customers in superexpress checkout = X2

E [ X1 -X2 ] = pij (X1 i - X2j)

= 0.06 * 0 + 0.06 * 1 + 0.05 * 2 + 0.00 * 3 + 0.00 * 4 + 0.05 * (-1) + 0.15 * 0 + 0.04 * 1 + 0.03 * 2 + 0.01 * 3 + 0.04 * (-2) + 0.05 * (-1) + 0.10 * 0 + 0.04 * 1 + 0.05 * 2 + 0.00 *(-3) + 0.06 * (-2) + 0.06 * (-1) + 0.07 * 0 + 0.06 * 1

= 0.13

E[ X1 - X2 ] = 0.13

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