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Unbiased Estimators

An estimatorfora statistical parameterissaid to be unbiased if

The mean of a sampleisan unbiased estimator of the population meanThiscan be used in a less obvious way in the following example.

A bag has 500 beans of twocolours, black and white. A sample of twenty beads is randomlyselected, with each bead put back in the bag immedicately afterselection, and the number of white beads is counted. Suppose we countn white beads is this sample. The proportion of white beads in thesample is then

To estimate the actualnumber of white beads, w, in the bag, calculate

The probability of pickinga white bead from the bag is sopicking a sample of twenty beads, replacing each one in the bag afterselection, gives a binomial probability distribution for the numberof white beads of

Thus the average number ofwhite beads is a sample is

The estimated number ofwhite beads in the bag using our single sample isso

soisan unbiased estimater for the mean.

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