next up previous index
Next: Example with a Uniform Up: Conditional Probability Previous: Beta Random Variable 10/28

Order Statistics 10/30

$X_1,X_2,\ldots,X_n$ are n iid continuous random variables with a common density f, a distribution function F.

Define:

\begin{displaymath}X_{(1)}=min(X_1,X_2,\ldots,X_n)\end{displaymath}


\begin{displaymath}X_{(2)}=min\{\{(X_1,X_2,\ldots,X_n)\} -\{X_{(1)}\}\}\end{displaymath}


\begin{displaymath}X_{(n)}=max(X_1,X_2,\ldots,X_n)\end{displaymath}

The ordered values of the iid sample are known as the order statistics.

The question we will try to reply to is:
What is the formula for the density of the kth order statistic?

We start with the extremes that are easy to handle:

First method

\begin{eqnarray*}F_{max}(x)=F_{(n)}(x) &=&
P(X_1\leq x, X_2\leq x, \ldots, X_n\l...
...& F^n(x)\\
f_{max}(x)=\frac{d}{dx}(F(x))^n=n(F(x))^{n-1}f(x)\\
\end{eqnarray*}



\begin{eqnarray*}F_{min}(x)=F_{(1)}(x) &=&
1-P(X_{min}>x)\\
&=&
1-P(X_1> x, X_2...
...\\
f_{min}(x) &=&-\frac{d}{dx}(1-F(x))^n=n(1-F(x))^{n-1}f(x)\\
\end{eqnarray*}


Second Method

\begin{eqnarray*}f_{(n)}(x)dx &=& P(X_{(n)}\in dx)\\
&=& P(\mbox{ one of the }X...
...n dx)P(\mbox{ all others } < x)\\
&=& n f(x) dx (F(x))^{n-1}\\
\end{eqnarray*}



\begin{eqnarray*}f_{(1)}(x)dx &=&P(X_{(1)}\in dx)\\
&=& P(\mbox{ one of the }X'...
...dx)P(\mbox{ all others } > x)\\
&=& n f(x) dx (1-F(x))^{n-1}\\
\end{eqnarray*}


The advantage of this method is that it can be generalized: X(k) is the k smallest of $X_1,X_2,\ldots,X_n$.


\begin{eqnarray*}f_{(k)}(x)dx &=&P(X_{(k)}\in dx)\\
&=& P(\mbox{ one of the }X'...
...)}(x)&=&n f(x)\binom{n-1}{k-1} (F(x))^{(k-1)}(1-F(x))^{(n-k)}\\
\end{eqnarray*}




 

Susan Holmes
1998-12-07