Concept explainers
Refer to Exercise 9.
- a. Find the conditional
probability massfunction pY|X(y | 4). - b. Find the conditional probability mass function pX|Y(x | 3).
- c. Find the conditional expectation E(Y | X =4).
- d. Find the conditional expectation E(X | Y = 3).
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).
a.
Find the conditional probability mass function
Answer to Problem 11E
The conditional probability mass function
y | |
0 | 0 |
1 | 0 |
2 | |
3 | |
4 |
Explanation of Solution
Given info:
In a lot of 500 bolts, X be the number that are downgraded and Y be the number that are scrap.
Calculation:
The joint probability mass function of X and Y their totals are given in the following table:
Y | ||||||
X | 0 | 1 | 2 | 3 | 4 | Total |
0 | 0.06 | 0.03 | 0.01 | 0.00 | 0.00 | 0.10 |
1 | 0.06 | 0.08 | 0.04 | 0.02 | 0.00 | 0.20 |
2 | 0.04 | 0.05 | 0.12 | 0.06 | 0.03 | 0.30 |
3 | 0.00 | 0.03 | 0.07 | 0.09 | 0.06 | 0.25 |
4 | 0.00 | 0.00 | 0.02 | 0.06 | 0.07 | 0.15 |
Total | 0.16 | 0.19 | 0.26 | 0.23 | 0.16 | 1 |
Table 1
The formula for finding the conditional probability mass function can be written as,
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
Thus, the conditional probability mass function
y | |
0 | 0 |
1 | 0 |
2 | |
3 | |
4 |
b.
Find the conditional probability mass function
Answer to Problem 11E
The conditional probability mass function
x | |
0 | 0 |
1 | |
2 | |
3 | |
4 |
Explanation of Solution
Calculation:
The formula for finding the conditional probability mass function can be written as,
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
For
The conditional probability mass function for
Substitute
Thus, the conditional probability mass function
x | |
0 | 0 |
1 | |
2 | |
3 | |
4 |
c.
Find the conditional expectation
Answer to Problem 11E
The conditional expectation
Explanation of Solution
Calculation:
The formula for finding the conditional expectation
Thus, the conditional expectation
d.
Find the conditional expectation
Answer to Problem 11E
The conditional expectation
Explanation of Solution
Calculation:
The formula for finding the conditional expectation
Thus, the conditional expectation
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Chapter 2 Solutions
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