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Line 23: === Sample mean of a normal random variable === Suppose one has a sequence of observations {''X''<sub>1</sub>, ''X''<sub>2</sub>, …} from a [[Normal distribution\|normal ''N''(''μ'', ''σ''<sup>2</sup>)]] distribution. To estimate ''μ'' based on the first ''n'' observations, one can use the [[sample mean]]: ''T<sub>n</sub>'' = (''X''<sub>1</sub> + … + ''X<sub>n</sub>'')/''n''. This defines a sequence of estimators, indexed by the sample size ''n''. From the properties of the normal distribution, we know the [[sampling distribution]] of this statistic: ''T''<sub>''n''</sub> is itself normally distributed, with mean ''μ'' and variance ''σ''<sup>2</sup>/''n''. Equivalently, <math style="vertical-align:-.3em">\scriptstyle (T_n-\mu)/(\sigma/\sqrt{n})</math> has a standard normal distribution:

Consistent estimator: Difference between revisions