Let X1, X2, ., X, be independent continuous random variables with common density function f Let X,, denote the ith smallest of X, .., X.

(a) Note that in order for X,, to equal x, exactly i - 1 of the X's must be less than x, one must equal x, and the other ni must all be greater than x Using this fact argue that the density function of X(i) is given by fx, (x) = n! (i-1)'(n-i) (F(x))''(F(x))"f(x).

(b) X() will be less than x if, and only if, how many of the X's are less than x

(c) Use

(b) to obtain an expression for P{X(,) x}

(d) Using

(a) and

(c) establish the identity (2) n! yk(1 y)" = - o (i 1)'(n i)' *'(1 x)*' dx for 0 y 1

(e) Let S, denote the time of the ith event of the Poisson process {N(t). t 0}. Find E[S,| N(t) = n] = ={ in i>n