Refer to Chebyshev’s inequality given in Exercise 44. Calculate P( | X - μ | ≥ kσ) for k = 2 and k = 3 when X ∼ Bin (20,. 5) , and compare to the corresponding upper bound. Repeat for X ∼ Bin (20,. 75)
Reference exercise 44
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 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/ 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
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