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!
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!
$\displaystyle \hat P\to p$ 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.