Bernoulli parameter expectation

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January 19th 2011, 12:16 PM
reflex

Bernoulli parameter expectation

I'm looking at this page: Bernoulli Distribution -- from Wolfram MathWorld

I think I'm missing something simple, but how do you get from equation 24 to 25?

Thanks!
January 19th 2011, 12:48 PM
emakarov

Yes, it looks a little... nonobvious.

The easiest thing is to immediately apply the Binomial theorem: $\displaystyle p\sum_{n=0}^N{N\choose n}p^n(1-p)^{N-n}=p(p+1-p)^N=p$ . Maybe they factored $(1-p)^N$ first: $\displaystyle\sum_{n=0}^N{N\choose n}p^n(1-p)^{N-n}=(1-p)^N\sum{N\choose n}\left(\frac{p}{1-p}\right)^n=(1-p)^N\left(1+\frac{p}{1-p}\right)^N=$ $\displaystyle(1-p)^N\frac{1}{(1-p)^N}=1$ .
January 19th 2011, 04:37 PM
matheagle

yes, just pull out the p, the rest sums to one.
January 21st 2011, 11:25 AM
reflex

Thanks!

Since it's an expectation of p-hat, shouldn't the first p (or all the p's) be a p-hat term?
January 21st 2011, 11:44 AM
matheagle

P is a constant, usually unknow, that's why we estimate it with a random variable... p-hat

$E(\hat p)=p$ and we like that p-hat is unbiased for p.

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