= n ( n – ) p { ∑ q n x } x = + np = n ( n – ) p ( q + p ) ( n – ) + np = n ( n – ) p + np ∴ Variance = E ( X ) – { E ( X )} = n p – np + np – n p = np ( – p ) = npq Hence, mean of the BD is np and the Variance is npq . Mean and variance in terms of raw moments and central moments are denoted as μ ′ , and μ respectively. Note Properties of Binomial distribution . Binomial distribution is symmetrical if p = q = .
. It is skew symmetric if p ≠ q . It is positively skewed if p < . and it is negatively skewed if p > .
. For Binomial distribution, variance is less than mean Variance npq = ( np ) q < np Example . A and B play a game in which their chance of winning are in the ratio : Find A ’s chance of winning atleast three games out of five games played. Let ‘ p ’ be the probability that ‘ A ’ wins the game.
Then we are given n = , p = / , q = – = (since q = – p ) Hence by binomial probability law, the probability that out of the games played, A wins ‘ x ’ games is given by P ( X = x ) = p x Cx