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Unformatted text preview: Lecture 16 16.1 Fisher and Student distributions. Consider X 1 , . . . , X k and Y 1 , . . . , Y m all independent standard normal r.v. Definition: Distribution of the random variable Z = X 2 1 + . . . + X 2 k Y 2 1 + . . . + Y 2 m is called Fisher distribution with degree of freedom k and m , and it is denoted as F k,m . Let us compute the p.d.f. of Z . By definition, the random variables X = X 2 1 + . . . + X 2 k 2 k and Y = Y 2 1 + . . . + Y 2 m 2 m have 2 distribution with k and m degrees of freedom correspondingly. Recall that 2 k distribution is the same as gamma distribution k 2 , 1 2 which means that we know the p.d.f. of X and Y : X has p.d.f. f ( x ) = ( 1 2 ) k 2 ( k 2 ) x k 2- 1 e- 1 2 x and Y has p.d.f. g ( y ) = ( 1 2 ) m 2 ( m 2 ) y m 2- 1 e- 1 2 y , for x 0 and y . To find the p.d.f of the ratio X Y , let us first recall how to write its cumulative distribution function. Since X and Y are always positive, their ratio is also positive and, therefore, for...
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Winter '09 term at MIT.
- Winter '09