Question

A certain market has both an express checkout line and a super-express checkout line. Let X1...

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.

 x2 0 1 2 3 x1 0 0.08 0.06 0.04 0.00 1 0.04 0.18 0.05 0.04 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

The difference between the number of customers in line at the express checkout and the number in line at the superexpress checkout is X1X2. Calculate the expected difference.

for above joijnt distribution:

 x2 x1 0 1 2 3 Total 0 0.0800 0.0600 0.0400 0.0000 0.1800 1 0.0400 0.1800 0.0500 0.0400 0.3100 2 0.0500 0.0400 0.1000 0.0600 0.2500 3 0.0000 0.0300 0.0400 0.0700 0.1400 4 0.0000 0.0100 0.0500 0.0600 0.1200 Total 0.1700 0.3200 0.2800 0.2300 1.0000

E(x1)=x1*P(x1)=0*0.18+1*0.31+2*0.25+3*0.14+4*0.12=1.71

E(X2)=1.57

here as

expected difference E(X1-X2)=E(X1)-E(X2)=1.71-1.57= 0.14