Binomial distribution:
Let X follow a binomial distribution with parameters n = 4 and p = 0.5. Tabulate P ( ¦X - μ¦ ≥ k ) and σ^{2}/k^{2} for k = 1,2,3,4,5 and verify that Chebyshev's lemma is true for each k. Comment on the usefulness of σ^{2} /k^{2} as an approximation of P ( ¦ X - μ ¦ ≥ k ).
Solution:
We have
Μ = np = 2, σ^{2 } =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 σ^{2}/k^{2} - is not a good approximation to the probability.