## ABSTRACT

Let X1, X2, …, Xn be n independent and identically distributed random variables, each with a probability density function f(x) and a cumulative distribution function F(x), i.e.,

{ }Pr ( ) ( )

iX x F x f t dt −∞

≤ = = ∫ For given X1, X2, …, Xn, the order statistics X(1), X(2), …, X(n) are random variables with their values arranged in ascending order, i.e.,

(1) (2) ( )nX X X≤ ≤ ≤

In particular, the random variables X(1) and X(n) are the smallest and largest order statistics, respectively, i.e.,

( )(1) 1 2min , , , nX X X X= …

( )( ) 1 2max , , ,n nX X X X= … In general, the random variable X(k) is known as the kth-order statistic. Let

fk(x) be the probability density function of X(k) Fk(x) be the cumulative distribution function of X(k)

Then fk(x) is given by

{ } { }1!( ) ( ) 1 ( ) ( )

( 1)!( )! k n k

k n

f x F x F x f x k n k

= −

− − (14.1)

and Fk(x) is given by

⎛ ⎞ = −⎜ ⎟⎝ ⎠∑( ) ( ) 1 ( )

n F x F x F x

i

(14.2)

where

! !( )!