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Answer: A result called Chebyshev’s inequality states that
Chapter 3, Problem 44E(choose chapter or problem)
Problem 44E
A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, P( | X - µ | k σ) ≤ 1/k2 . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.
a. What is the value of the upper bound for k = 2? K + 3? K = 4? K= 5? K = 10?
b. Compute µ and σ for the distribution of Exercise 13. Then evaluate P(|X - µ| ≥ kσ) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability?
c. Let X have possible values -1, 0, and 1, with probabilities 1/18, 8/9 and 1/8 , respectively. What is P(|X - µ|≥ 3σ), and how does it compare to the corresponding bound?
d. Give a distribution for which P(|X - µ|≥ 5σ) = .04.
Reference exercise -13
A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.
Calculate the probability of each of the following events.
a. {at most three lines are in use}
b. {fewer than three lines are in use}
c. {at least three lines are in use}
d. {between two and five lines, inclusive, are in use}
e. {between two and four lines, inclusive, are not in use}
f. {at least four lines are not in use}
Questions & Answers
QUESTION:
Problem 44E
A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, P( | X - µ | k σ) ≤ 1/k2 . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/k2.
a. What is the value of the upper bound for k = 2? K + 3? K = 4? K= 5? K = 10?
b. Compute µ and σ for the distribution of Exercise 13. Then evaluate P(|X - µ| ≥ kσ) for the values of k given in part (a). What does this suggest about the upper bound relative to the corresponding probability?
c. Let X have possible values -1, 0, and 1, with probabilities 1/18, 8/9 and 1/8 , respectively. What is P(|X - µ|≥ 3σ), and how does it compare to the corresponding bound?
d. Give a distribution for which P(|X - µ|≥ 5σ) = .04.
Reference exercise -13
A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.
Calculate the probability of each of the following events.
a. {at most three lines are in use}
b. {fewer than three lines are in use}
c. {at least three lines are in use}
d. {between two and five lines, inclusive, are in use}
e. {between two and four lines, inclusive, are not in use}
f. {at least four lines are not in use}
ANSWER:
Answer :
Step 1 of 4 :
Given, A result called Chebyshev’s inequality states that for any probability distribution of an rv X and any number k that is at least 1, P( | X - µ | k σ) ≤ 1/k2 . In words, the probability that the value of X lies at least k standard deviations from its mean is at most 1/.
a)
The claim is to find the upper bound for k = 2, k = 3, k = 4, k = 5 and k = 10.
This is a simple problem of evaluating 1 for each given k.
For k = 2 , = 0.25
k = 3 , =0.11
k = 4 , =0.0625
k = 5 , = 0.04
k = 10 , = 0.01