Refer to Exercise 9.

a. Find the mean of the total number of unacceptable bolts (those that are classified as either downgraded or scrap).

b. Find the variance of the total number of unacceptable bolts.

c. Find the probability that the total number of unacceptable bolts is exactly 5.

EXERCISE 9: Bolts manufactured for a certain purpose may be classified as acceptable (suitable for the intended purpose), downgraded (unsuitable for the intended purpose but acceptable for a different purpose), or scrap (unsuitable for any purpose). In a lot of 500 bolts, let X be the number that are downgraded and let Y be the number that are scrap. Assume that the joint probability mass function of X and Y is given in the following table.

a. Find the marginal probability mass function of X.

b. Find the marginal probability mass function of Y.

c. Are X and Y independent? Explain.

d. Find μX and μY.

e. Find σX and σY.

f. Find Cov(X, Y).

g. Find ρ(X, Y).

Answer

Step 1 of 7</p>

Here given the information about downgraded and scrap bolts

We need to find the mean and variances of unacceptable bolts

Let X is the no. of downgraded bolts

Let Y is the no. of scrap bolts

Total no. of unacceptable bolts= X+Y

The joint probability mass function is

X\ Y |
0 |
1 |
2 |
3 |
4 |
PX(X) |

0 |
0.06 |
0.03 |
0.01 |
0.0 |
0.0 |
0.1 |

1 |
0.06 |
0.08 |
0.04 |
0.02 |
0.0 |
0.2 |

2 |
0.04 |
0.05 |
0.12 |
0.06 |
0.03 |
0.3 |

3 |
0.0 |
0.03 |
0.07 |
0.09 |
0.06 |
0.25 |

4 |
0.0 |
0.0 |
0.02 |
0.06 |
0.07 |
0.15 |

Py(Y) |
0.16 |
0.19 |
0.26 |
0.23 |
0.16 |
1 |

Step 2 of 7</p>

The marginal probability mass function of X is PX(X)=

X |
0 |
1 |
2 |
3 |
4 |
Total |

PX(X) |
0.1 |
0.2 |
0.3 |
0.25 |
0.15 |
1 |

The marginal probability mass function of Y is PY(Y)=

Y |
0 |
1 |
2 |
3 |
4 |
Total |

PY(Y) |
0.16 |
0.19 |
0.26 |
0.23 |
0.16 |
1 |