## Consistent Estimators

Given a sample from apopulation, we can estimate the mean of the population from As the sample sizeincreases, the variance of decreases.This property makes a useful estimator for the population mean, sinceby increasing the sample size n, we can reduce the variance of If isalso an unbiased estimator for the population mean then isa consistent estimator for If an estimator fora population parameter hasthe properties and asthe sample size for calculating tendsto infinity, then isa consistent estimator for The sample mean isan unbiased estimator for the population mean since Since also, if the variance of the population is andthe mean is found from a sample of size n using then and isa consistent estimator for Similarly, for a binomialdistribution with proportion fromwhich a sample of size istaken, and we record the number of success as andthe proportion as theproportion of 'successes' is expected to be The variance of is The variance of tendsto zero as n tends to infinity so isa consistent estimator for 