Refer to Exercise 12. Assume that the cost of an engine repair is $50, and the cost of a transmission repair is $100. Let T represent the total cost of repairs during a one-hour time interval.

a. Find μT.

b. Find σT.

c. Find P(T = 250).

Exercise 12

Automobile engines and transmissions are produced on assembly lines, and are inspected for defects after they come off their assembly lines. Those with defects are repaired. Let X represent the number of engines, and Y the number of transmissions that require repairs in a one-hour time interval. The joint probability mass function of X and Y is as follows:

a. Find the marginal probability mass function px(x).

b. Find the marginal probability mass function py(y).

c. Find μx.

d. Find μy.

e. Find σx.

f. Find σy.

g. Find Cov(X,Y).

h. Find ρx,y.

Solution 14E

Step1 of 4:

We have the cost of an engine repair is $50, and the cost of a transmission repair is $100.

Let us take T it represent the total cost of repairs during a one-hour time interval.

That is

y | ||||

x | 0 | 1 | 2 | 3 |

0 | 0.13 | 0.1 | 0.07 | 0.03 |

1 | 0.12 | 0.16 | 0.08 | 0.04 |

2 | 0.02 | 0.06 | 0.08 | 0.04 |

3 | 0.01 | 0.02 | 0.02 | 0.02 |

Here our goal is:

a).We need to find .

b).We need to find .

c).We need to find P(T = 250).

Step2 of 4:

a).

Consider,

y | Total | ||||

x | 0 | 1 | 2 | 3 | |

0 | 0.13 | 0.1 | 0.07 | 0.03 | 0.33 |

1 | 0.12 | 0.16 | 0.08 | 0.04 | 0.4 |

2 | 0.02 | 0.06 | 0.08 | 0.04 | 0.2 |

3 | 0.01 | 0.02 | 0.02 | 0.02 | 0.07 |

Total | 0.28 | 0.34 | 0.25 | 0.13 | 1 |

Where

,