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With any unbiased estimator, why is the distribution of the estimates also the distribution of the statistic?

Eg, suppose we have 5 independent and identically distributed normal random variables with variance 1 and mean (unknown parameter).

We observe some numbers say { 4, 5, -2 ,7 , 12}.

and we use sample mean as the estimator for mean. The sample mean is clearly normally distributed.

But why is this also the distribution for mean