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| > kσ) < 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 m and s 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 1/18, 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.