Property A: The moment generating function for a random variable with distribution B(n, p) is
where q = 1 – p.
Proof: Using the definition of the binomial distribution and the definition of a moment generating function, we have
Observation: You can use the moment generating function to calculate the mean and variance (namely Property 1 of Binomial Distribution).
Theorem 1: If x is a random variable with distribution B(n, p), then for sufficiently large n, the distribution of the variable
Proof: By the linear transformation properties of the moment generating function (i.e Property 3 of General Properties of Distributions),
Taking the natural log of both sides, and then expanding the power series of we get
Since p + q = 1 we have
If n is made sufficiently large can be made large enough that for any fixed θ the absolute value of the sum above will be less than 1. Let
Thus for sufficiently large n, |z| < 1. The ln term in the previous expression is ln(1+z) where |z|<1, and so we may expand this term as follows:
Rearranging the terms, we have
for values of ck which don’t involve n, σ or θ. Since μ = np and σ2 = np(1 – p), the coefficient of the θ term is 0 and the coefficient of the θ2 term is 1. Thus,
Since the coefficient of each term in the sum has form
But note that by Property 3 of Normal Distribution the moment generating function for a random variable z with distribution N(0, 1) is
The result now follows by Corollary 1 of General Properties of Distributions.