So in a binomial distribution, we know that p^ = X / n is an unbiased estomator for p. Which means that when n approaces infinity, p^ approaches p. But how can I prove this mathematically? Any help would be appreciated!

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- Apr 30th 2010, 04:11 PMgralla55Unbiased estimator in binomial distribution
So in a binomial distribution, we know that p^ = X / n is an unbiased estomator for p. Which means that when n approaces infinity, p^ approaches p. But how can I prove this mathematically? Any help would be appreciated!

- Apr 30th 2010, 06:12 PMGnomeSain
- Apr 30th 2010, 07:31 PMgralla55
Yes of course, I wrote it a little unprecise. But intuitively it's easy to see that the expected value of p^ has to be equal to p. Since p, the probability, is defined as the mean of an unlimited population. I'm looking for a more formal proof though.

- Apr 30th 2010, 09:23 PMmatheagle
$\displaystyle \hat P\to p$ almost surely by the strong law of large numbers

$\displaystyle \hat P={Y\over n}={\sum_{i=1}^n X_i\over n}\to p$

where Y is Binomial(n,p) and $\displaystyle X_i$ are iid Bernoulli's with mean p.

You can always use chebyshev's inequality and obtain convergence in probability.