Binomial Distribution – Advanced

Property 0: B(n, p) is a valid probability distribution

Proof: the main thing that needs to be proven is that

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where f(x) is the pdf of B(n, p). This follows from the well-known Binomial Theorem since

image9086

The Binomial Theorem that

image9087

can be proven by induction on n.

Property 1

image506 image507

Proof (mean): First we observe

Now

image3242

where m = n − 1 and i = k − 1 . But

where fm,p(i) is the pdf for B(m, p), and so we conclude μ = E[x] = np.

Proof (variance):

We begin using the same approach as in the proof of the mean:

image3248

image3252

Thus,
image3253

2 Responses to Binomial Distribution – Advanced

  1. Robert says:

    Two points in the variance proof are not clear. Why does the probability function sum up to one when the parameter C(m,i+1) is used instead of C(m,i) as in:
    SUM(i=0,m){C(m,i+1)p^i(1-p)^m-i
    Second, Why does the following sum equal to mp when the “i” is part of the multiplicand?
    SUM(i=0,m){i*C(m,i+1)p^i(1-p)^m-i}. I am not familiar with this identity. Thanks!

    • Charles says:

      Robert,
      The proofs that were given were not quite right. I have now made a some corrections which should address the issues that you raised. Thanks for bringing these to light.
      Charles

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