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 day, and let *X*_{2} denote the number of
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.

x_{2} |
|||||

0 | 1 | 2 | 3 | ||

x_{1} |
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 *X*_{1} − *X*_{2}.
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**

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