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Example with a Uniform Random Variable

The density of a uniform random variable is

\begin{displaymath}f(x)=1, 0<x<1, \qquad f(x)=0\mbox{ otherwise}\end{displaymath}

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

\begin{displaymath}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.

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

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

\begin{displaymath}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)

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

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


Susan Holmes