# Unbiased estimator in binomial distribution

• Apr 30th 2010, 04:11 PM
gralla55
Unbiased 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 PM
GnomeSain
Quote:

Originally Posted by gralla55
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!

An estimator is unbiased if its expected value is equal to the parameter. The limit criteria you described means an estimator is consistent.
• Apr 30th 2010, 07:31 PM
gralla55
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 PM
matheagle
$\displaystyle \hat P\to p$ 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.