Refer to Exercise 1.

a. Find the conditional probability mass function PY|X(y|0).

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

c. Find the conditional expectation E(Y | X = 0).

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

Step 1 of 4:

The number of days on which the ozone standard is exceeded is X and

The number of days on which the particulate matter standard is exceeded is Y.

Then the joint probability mass function of X and Y is given in the table below.

Y |
|||

X |
0 |
1 |
2 |

0 |
0.10 |
0.11 |
0.05 |

1 |
0.17 |
0.23 |
0.08 |

2 |
0.06 |
0.14 |
0.06 |

Our goal is:

a). Find the conditional probability mass function .

b). Find the conditional probability mass function .

c). Find the conditional expectation .

d). Find the conditional expectation .

a).

Now we have to find the conditional probability mass function .

We have a table.

Y |
|||

X |
0 |
1 |
2 |

0 |
0.10 |
0.11 |
0.05 |

1 |
0.17 |
0.23 |
0.08 |

2 |
0.06 |
0.14 |
0.06 |

Then the conditional probability mass function is

and

Then, we substitute and values.

Therefore is 0.3846.

and

Then, we substitute and values.

Therefore is 0.4230.

and

Then, we substitute and values.

Therefore is 0.1923.

Hence the conditional probability mass function is

.