A certain market has both an express checkout line and a express checkout line.

Question

A certain market has both an express checkout line and a express checkout line. Let X_1 denote the number of the express checkout at a particular time of day, and let X_2 denote the number of customers in line customers in line at the superexpress checkout at the same time. Suppose the joint pmf of 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

X1 - X2 takes the value 0 along the central diagonal running from top left to bottom right, value of 1 along the 2 neighbouring diagonals of the central diagonal, and value of 2 along the 2 neighbouring diagonals of the provious pair and so on.

PDF of X1 - X2:

E(X1-X2) = 0*0.43 + 1*0.34 + 2*0.22 + 3*0.01 + 4*0 = 0.81

X1-X2 0 1 2 3 4 Probability 0.08+0.18+0.10+0.07 = 0.43 0.04+0.04+0.04+0.06+0.05+0.05+0.06 = 0.34 =0.04+0.05+0.05+0.03+0.05 = 0.22 0.01 0.00

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