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, . 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 ? ?

? ? ?

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 possible values , 0, and 1, with probabilities

, , and , respectively. What is , and how does it compare to the corresponding bound?

d. Give a distribution for which .