Concept explainers
The joint
- a. Determine the conditional pdf of Y given that X = x and the conditional pdf of X given that Y = y.
- b. If the pressure in the right tire is found to be 22 psi, what is the
probability that the left tire has a pressure of at least 25 psi? Compare this to P(Y ≥ 25). - c. If the pressure in the right tire is found to be 22 psi, what is the expected pressure in the left tire, and what is the standard deviation of pressure in this tire?
a.
Find the conditional pdf of Y given that
Answer to Problem 19E
The conditional pdf of Y given that
The conditional pdf of X given that
Explanation of Solution
Given info:
Denote X is the actual air pressure in the right tire of a vehicle and Y is that in the left tire. The joint pdf of X and Y is:
Calculation:
From Exercise 9, the marginal pdf’s of X and Y are found to be:
The conditional pdf of Y given that
The expression for the conditional pdf for Y given X is:
Thus, the conditional pdf of Y given that
Replace
Thus, the conditional pdf of Y given that
Again the conditional pdf of X given that
Thus, the conditional pdf of X given that
b.
Find the probability that the left tire has a pressure of at least 25 psi, if the pressure in the right tire is found to be 22 psi.
Compare the obtained probability to
Answer to Problem 19E
The probability that the left tire has a pressure of at least 25 psi, if the pressure in the right tire is found to be 22 psi is 0.5559.
The probability that the left tire has a pressure of at least 25 psi is slightly higher if the right tire already has a pressure of 22 psi.
Explanation of Solution
Calculation:
The probability that the left tire has a pressure of at least 25 psi, if the pressure in the right tire is found to be 22 psi is:
Now,
Thus, the probability that the left tire has a pressure of at least 25 psi is slightly higher if the right tire already has a pressure of 22 psi.
c.
Find the expected pressure in the left tire, if the pressure in the right tire is found to be 22 psi.
Find the standard deviation of pressure in the left tire under the condition.
Answer to Problem 19E
The expected pressure in the left tire, if the pressure in the right tire is found to be 22 psi is 25.373 psi.
The standard deviation of pressure in the left tire under the condition is 2.87 psi.
Explanation of Solution
Calculation:
The expected pressure in the left tire, if the pressure in the right tire is found to be 22 psi is:
The variance of pressure in the left tire under the condition is
Now,
Thus,
Hence, the standard deviation is:
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Chapter 5 Solutions
Probability and Statistics for Engineering and the Sciences
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw HillCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,