Binomial and Normal Distributions – Advanced

Property A: The moment generating function for a random variable with distribution B(n, p) is

image3254

where q = 1 – p.

Proof: Using the definition of the binomial distribution and the definition of a moment generating function, we have

image3256

Observation: You can use the moment generating function to calculate the mean and variance (namely Property 1 of Binomial Distribution).

image3257 image3258 image3259 image3260

Theorem 1: If x is a random variable with distribution B(n, p), then for sufficiently large n, the distribution of the variable

image522

where
image523

Proof: By the linear transformation properties of the moment generating function (i.e Property 3 of General Properties of Distributions),

image3266

Taking the natural log of both sides, and then expanding the power series of e^{\theta/\sigma} we get

image3268

Since p + q = 1 we have

image3270

If n is made sufficiently large \sigma = \sqrt{npq} can be made large enough that for any fixed θ the absolute value of the sum above will be less than 1. Let

image3272

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:

image3276This means that

image3277

Rearranging the terms, we have

image3278

for values of ck which don’t involve n, σ or θ. Since μ = np and σ2np(1 – p), the coefficient of the θ term is 0 and the coefficient of the θ2 term is 1. Thus,

image3285

Since the coefficient of each term in the sum has form

image3286

Thus
image3287

and so
image3288

But note that by Property 3 of Normal Distribution the moment generating function for a random variable z with distribution N(0, 1) is

image3290

The result now follows by Corollary 1 of General Properties of Distributions.

One Response to Binomial and Normal Distributions – Advanced

  1. Isanka says:

    Thank you soooo much

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