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# lec16 - Lecture 16 16.1 Fisher and Student distributions...

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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 χ 2 k and Y = Y 2 1 + . . . + Y 2 m χ 2 m have χ 2 distribution with k and m degrees of freedom correspondingly. Recall that χ 2 k distribution is the same as gamma distribution Γ 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 Γ( k 2 ) x k 2 - 1 e - 1 2 x and Y has p.d.f. g ( y ) = ( 1 2 ) m 2 Γ( m 2 ) y m 2 - 1 e - 1 2 y , for x 0 and y 0 . 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 t 0 we can write: X Y t = ( X tY ) = { I ( X tY ) } = Z 0 Z 0 I ( x ty ) f ( x ) g ( y ) dxdy = Z 0 Z ty 0 f ( x ) g ( y )

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