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.

		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 X₁ − X₂. Calculate the expected difference.

Answer 1

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