Example with a Uniform Random Variable

The density of a uniform random variable is

$$f(x)=1, 0<x<1, \qquad f(x)=0\mbox{ otherwise}$$

and the cumulative distribution function is the identity between 0 and 1.So that applying the above formula for the kth order statistic of n independent uniform random variables (0,1) gives the density

$$f_{(k)}(x)=n\binom{n-1}{k-1}x^{(k-1)}(1-x)^{(n-k)}\mbox{ between 0 and 1} \qquad \mbox{ and 0 elsewhere}$$

This is in fact a density we will encounter several times, it is the Beta(k,n-k+1) density.

Example:
Suppose five independent uniforms U₁, U₂...U₅.

Find the joint density of U₍₂₎ andU₍₄₎ :

$$P(U_{(2)} \in dx, U_{(4)} \in dy)$$

$$\begin{eqnarray*}&=& P \left( \mbox{one}\ U_i \mbox{in}\ (0, x), \mbox{one in} d... ...U_1 \in dy, U_5 \in (y, 1) \right) \\ &=& 5! x dx(y-x) dy (1-y) \end{eqnarray*}$$

$$f_{(2)(4)} (x, y) = \left\{ \begin{array}{ll} 5! & x(y-x)(1-y) \\ & 0<x<y<1\\ 0 & \mbox{elsewhere} \end{array}\right.$$

This is the density of which I showed you a picture in class when I defined joint densities, it is only non-zero for

0<x<y<1