# Thread: Order statistics - determine the cdf

1. ## Order statistics - determine the cdf

Let $u_1,...,u_N$ be N independent random variables uniformly distributed on the interval $[0,1]$. Determine the cdf of $y:= max_n u_n$ and $z := min_n u_n$

So we are after the cdf of $F_y(s)=P(max_u \leq s)$ which is the probability that $max_u$ is greater then $s_1, s_2...$? not sure how to formalise/what distribution this is?

Thanks for any help..

2. Originally Posted by Robb
Let $u_1,...,u_N$ be N independent random variables uniformly distributed on the interval $[0,1]$. Determine the cdf of $y:= max_n u_n$ and $z := min_n u_n$

So we are after the cdf of $F_y(s)=P(max_u \leq s)$ which is the probability that $max_u$ is greater then $s_1, s_2...$? not sure how to formalise/what distribution this is?

Thanks for any help..
Hint:

$max \{u_1, u_2, \dots , u_n\} \leq y$
if and only if
$u_i \leq y$ for all $i = 1, 2, \dots , n$.

3. Originally Posted by Robb
Let $u_1,...,u_N$ be N independent random variables uniformly distributed on the interval $[0,1]$. Determine the cdf of $y:= max_n u_n$ and $z := min_n u_n$

So we are after the cdf of $F_y(s)=P(max_u \leq s)$ which is the probability that $max_u$ is greater then $s_1, s_2...$? not sure how to formalise/what distribution this is?

Thanks for any help..
You will find detailed proofs in most mathematical statistics books, using Google and also by searching this subforum (use the Search tool).

4. hmm, so after doing a bit of research...
The distribution of independent order statistics would be;
For the Maximum
$F_{Y_{N}}(s)=P(Y_{n}\leq s)=P(Y_1\leq s)P(Y_2\leq s)...P(Y_n\leq s)=[F_U(s)]^N$
So since since it is the uniform distribution with $F_U(s)=s$ the cdf is $F_{Y_{N}}(s)=s^N$

And the minimum will be
$F_{Z_1}(s)=1-P(Z_1 > s)=1-[1-F_U(s)]^N$

5. Originally Posted by Robb
hmm, so after doing a bit of research...
The distribution of independent order statistics would be;
For the Maximum
$F_{Y_{N}}(s)=P(Y_{n}\leq s)=P(Y_1\leq s)P(Y_2\leq s)...P(Y_n\leq s)=[F_U(s)]^N$
So since since it is the uniform distribution with $F_U(s)=s$ the cdf is $F_{Y_{N}}(s)=s^N$

And the minimum will be
$F_{Z_1}(s)=1-P(Z_1 > s)=1-[1-F_U(s)]^N$
Order Statistics 10/30