# lec14 - Lecture 14 14.1 Estimates of parameters of normal...

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Lecture 14 14.1 Estimates of parameters of normal distribu- tion. Let us consider a sample X 1 , . . . , X n N ( α, σ 2 ) from normal distribution with mean α and variance σ 2 . Using different methods (for example, maximum likelihood) we showed that one can take ¯ X as an estimate of mean α and ¯ X 2 - ( ¯ X ) 2 as an estimate of variance σ 2 . The question is: how close are these estimates to actual values of unknown parameters? By LLN we know that these estimates converge to α and σ 2 , ¯ X α, ¯ X 2 - ( ¯ X ) 2 σ 2 , n → ∞ , but we will try to describe precisely how close ¯ X and ¯ X 2 - ( ¯ X ) 2 are to α and σ 2 . We will start by studying the following Question: What is the joint distribution of ( ¯ X, ¯ X 2 - ( ¯ X ) 2 ) when the sample X 1 , . . . , X n N (0 , 1) has standard normal distribution. Orthogonal transformations. The student well familiar with orthogonal transformations may skip to the be- ginning of next lecture. Right now we will repeat some very basic discussion from

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