Dynamic Pricing and Auction - Department of Industrial Engineering ...

Density of the k th lowest order statisticthDensity of k lowest:Pr{ x< x ≤ x+dx}( k )=Pr{[ P{[ x < one of the X ' s ≤ x+ dx ] ∩[ X ≤ x for ( k −1) X ' s]At limit when dx → 0 :i i i∩[ X > x for the remaining ( n−k) X ' s]}[ F( x+ dx) −F( x)]f v ) = c c [ F( x)] .[1 −F( x)]dxn!k−1n−kf ( v( )) = kf ( x)[ ).[ F ( x)] .[1 F ( x)]( k−1)!( n−k)!−n n−1 k−1n−k(( k ) 1 k −1ii