uniform random numbers and pdf and cdf

111 Views Asked by At

We say $U_1$, $U_2$ and $ U_3$ are independent uniform random numbers. We are arrange them, like $U_{(1)}$, $U_{(2)}$ and $U_{(3)}$ from small to big.

I have to determine what the pdf, cdf are. How can I find them if I don't know the interval where the numbers are chosen from? Can somebody explain me?

1

There are 1 best solutions below

3
On

First do it for $V_1,V_2,V_3$ with standard uniform distribution (so defined on $[0,1]$).

Then let $a,b$ be real numbers with $a<b$ and let $U_i:=a+(b-a)V_i$.

Then $U_i$ will have uniform distribution over $[a,b]$ and you can easily find that: $$U_{(i)}=a+(b-a)V_{(i)}\text{ for }i=1,2,3$$

These equalities will enable you to find CDF and PDF of the $U_{(i)}$.