A random variable X follows a binomial distribution B(n, p) . If the expectation of X^2 is 18 and P(X=3) = 2 P(X=2) , then the mean of the distribution is

Step-by-step solution

Given that P(X=3) = 2 P(X=2) . ^ n C₃ p^3 q^ n-3 = 2 ^ n C₂ p^2 q^ n-2 n(n-1)(n-2) 6 p = 2 n(n-1) 2 q n-2 6 p = q (n-2) p = 6 q Substituting q = 1 - p , we get: np - 2p = 6(1 - p) np - 2p = 6 - 6p np = 6 - 4p We are also given that E(X^2) = 18 . Using the relation E(X^2) = Var (X) + (E(X))^2 , we have: E(X^2) = npq + (np)^2 = np(1-p) + (np)^2 = np(1 - p + np) Substitute np = 6 - 4p into the equation: (6 - 4p)(1 - p + 6 - 4p) = 18 (6 - 4p)(7 - 5p) = 18 42 - 30p - 28p + 20p^2 = 18 20p^2 - 58p + 24 = 0 10p^2 - 29p + 1