Refer to Exercise 12.

a. Find the conditional probability mass function PY|X(y | 3).

b. Find the conditional probability mass function pX|Y(x|1).

c. Find the conditional expectation E(Y | X = 3).

d. Find the conditional expectation E(X | Y = 1).

Solution 15E

Step1 of 5:

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 the conditional probability mass function

b).We need to find the conditional probability mass function

c).We need to find the conditional expectation E(Y|X = 3).

d).We need to find the conditional expectation E(X|Y = 1).

Step2 of 5:

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

,