Chebyshev’s inequality, (see Exercise 44, Chapter 3), is

QUESTION:

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 – ? ?k?) ?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 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 | k ?) ? 1/?k?2 . In words, the probability that the value of ?X ?lies at least ?k ?standard deviations from its mean is at most 1/?k?2. a.? hat is the value of the upper bound for k = 2? K + 3? K = 4? K= 5? K = 10? b.? ?Compute µ and ? for the distribution of Exercise 13. Then evaluate P(|X - µ| ? k?) 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|? 3?), and how does it compare to the corresponding bound? d.? ?Give a distribution for which P(|X - µ|? 5?) = .04.