10_06 - n = 4 from a probability distribution with the...

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STAT 410 Examples for 10/06/2008 Fall 2008 In general, if X 1 , X 2 , … , X n is a random sample of size n from a continuous distribution with cumulative distribution function F ( x ) and probability density function f ( x ), then F max X i ( x ) = P ( max X i x ) = P ( X 1 x , X 2 x , … , X n x ) = P ( X 1 x ) P ( X 2 x ) P ( X n x ) = ( ) ( ) n x F . f max X i ( x ) = F ' max X i ( x ) = ( ) ( ) ( ) 1 F x f x n n - . 1 – F min X i ( x ) = P ( min X i > x ) = P ( X 1 > x , X 2 > x , … , X n > x ) = P ( X 1 > x ) P ( X 2 > x ) P ( X n > x ) = ( ) ( ) n x F 1 - . F min X i ( x ) = ( ) ( ) n x F 1 1 - - . f min X i ( x ) = F ' min X i ( x ) = ( ) ( ) ( ) 1 F 1 x f x n n - - . Let Y k = k th smallest of X 1 , X 2 , … , X n . F Y k ( x ) = P ( Y k x ) = P ( k th smallest observation x ) = P ( at least k observations are x ) = ( ) ( ) ( ) ( ) ° = - - ± ± ² ³ ´ ´ µ n k i i n i x x i n F 1 F . f Y k ( x ) = F ' Y k ( x ) = ( ) ( ) ( ) ( ) ( ) ( ) ( ) F 1 F 1 1 ! ! ! x f x x k n k n k n k - - - - - .
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1. Let X 1 , X 2 , X 3 , X 4 be a random sample ( i.i.d. ) of size n = 4 from a probability distribution with the p.d.f. f ( x ) = 3 / x 4 , x > 1. Let Y
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Unformatted text preview: n = 4 from a probability distribution with the p.d.f. f ( x ) = 3 / x 4 , x > 1. Let Y k = k th smallest of X 1 , X 2 , … , X n . a) Find P ( Y 4 < 1.75 ) = P ( max X i < 1.75 ). b) Find P ( Y 4 > 2 ) = P ( max X i > 2 ). c) Find P ( Y 1 > 1.25 ) = P ( min X i > 1.25 ). d) Find P ( 1.1 < Y 1 < 1.2 ) = P ( 1.1 < min X i < 1.2 ). e) Find P ( 1.1 < Y 2 < 1.2 ). 2. Let X 1 , X 2 , … , X n be a random sample ( i.i.d. ) from Uniform ( , a ) probability distribution. Let Y k = k th smallest of X 1 , X 2 , … , X n . Find E ( Y k ). 3. Let X i be an Exponential ( λ i ) random variable, i = 1, 2, … , n . Suppose X 1 , X 2 , … , X n are independent. Find the probability distribution of min X i ....
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