# lec6 - Lecture 6 Let us compute Fisher information for some...

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Let us compute Fisher information for some particular distributions. Example 1. The family of Bernoulli distributions B ( p ) has p.f. f ( x | p ) = p x (1 - p ) 1 - x and taking the logarithm log f ( x | p ) = x log p + (1 - x ) log(1 - p ) . The second derivative with respect to parameter p is ∂p log f ( x | p ) = x p - 1 - x 1 - p , 2 ∂p 2 log f ( x | p ) = - x p 2 - 1 - x (1 - p ) 2 then we showed that Fisher information can be computed as: I ( p ) = - 2 ∂p 2 log f ( X | p ) = X p 2 + 1 - X (1 - p ) 2 = p p 2 + 1 - p (1 - p ) 2 = 1 p (1 - p ) . The MLE of p is ˆ p = ¯ X and the asymptotic normality result from last lecture becomes n p - p 0 ) N (0 , p 0 (1 - p 0 )) which, of course, also follows directly from the CLT. Example. The family of exponential distributions E ( α ) has p.d.f. f ( x | α ) = ± αe - αx , x 0 0 , x < 0 and, therefore, log f ( x | α ) = log α - αx 2 ∂α 2 log f ( x | α ) = - 1 α 2 . 24

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## This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.

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lec6 - Lecture 6 Let us compute Fisher information for some...

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