Binomial distribution:

Let X follow a binomial distribution with parameters n = 4 and p = 0.5. Tabulate P ( ¦X - μ¦ ≥ k ) and σ²/k² for k = 1,2,3,4,5 and verify that Chebyshev's lemma is true for each k. Comment on the usefulness of σ² /k² as an approximation of  P ( ¦ X - μ ¦ ≥ k ).

Solution:

We have

Μ = np = 2, σ²  =npq = 1 and p(X =x) = (.5)⁴ /16,

X= 0,1,2,3,4.

Therefore we obtain, in a tabular form

Chebyshev's lemma is clearly satisfied for every k. However, the upper bound to the probability is weak and the estimate σ²/k² - is not a good approximation to the probability.