Chebyshev’s inequality, (see Exercise 44, Chapter 3), is valid for continuous as well as discrete distributions. It states that for any number k satisfying k ≥ 1,P(|X – μ ≥kσ) ≤1/k2 (see Exercise 44 in Chapter 3 for an interpretation). Obtain this probability in the case of a normal distribution for , 2, and 3, and compare to the upper bound.
Reference exercise 44A 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.
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