We have a weighted coin that has probability of flipping hea

We have a weighted coin that has probability of flipping heads up as p = 0.05. Let X be the Binomial(N, p) random variable describing the number of heads-up flips in N flips with a coin with probability of heads p. Compute E[X^3] by computing E[X(X - 1 )(X - 2)] and using the fact that Var(X) = N_p{ 1 - p) and E(X) = N_p.

Solution

E[X3]=E[X(X-1)(X-2)

=E[X]E[X-1]E[X-2]

=np*np*np

=n3p3