43. 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, 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 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 three possible values, 1, 0, and 1, with probabilities , , and , respectively. What is , and how does it compare to the corresponding bound?

d. Give a distribution for which 5s2  .04.