Question

Let $X_{\left(1\right)},X_{\left(2\right)},...,X_{\left(n\right)}$be the order statistics of a set of n independent uniform $(0,1)$random variables. Find the conditional distribution of $X_{\left(n\right)}$ given that$X_{\left(1\right)=S_1},X_{\left(2\right)=S_2},....,X_{(n-1)=S(n-1)}$.

Short answer

Conditional distribution is$X_{\left(n\right)}=f(x_{n}0<x_{n-1}<1)$

Step-by-step solution

Conditional distribution :

If X and Y are two jointly distributed random variables, the conditional distribution of Y given X is the probability distribution of Y when X has a known value.

Explanation : 

Let,

$f\left(\frac{x_n}{x_1....x_{(n-1)}}\right)=\frac{f(x_n,f(x_1),...(x_{n-1}\left)\right)}{f\left(x_1\right)....f\left(x_{n-1}\right)}$

By definition of $x_{1}and{x}_2$.

$$\begin{gathered} f\left(\frac{x_n}{x_1....x_{(n-1)}}\right)=f\left(x_n\right) \\ f\left(\frac{x_n}{x_1....x_{(n-1)}}\right)=f\left(x_n\right)....0<x_{\left(n\right)}<1 \\ f\left(\frac{x_n}{x_1....x_{(n-1)}}\right)=f\left(x_n\right)...x_{(n-1)} \\ f\left(\frac{x_n}{x_1....x_{(n-1)}}\right)=f(x_{n}0<x_{n-1}<1) \end{gathered}$$

A uniform distribution has a partial differential function and a continuous differential function.

$$\begin{gathered} f\left(x\right)=1:0<x<1 \\ f\left(x\right)=x:0<x<1\left[f\right(x_n)=n[f{\left(x\right)}^{n-1}.f\left(x\right)\left]\right] \\ f\left(x_{\left(n\right)}\right)=n{\left[x\right]}^{n-1}:0<x<1 \\ f(x_{\left(n\right)}/x_{\left(1\right)}.....x_{(n-1)}):n{x}^{n-1}:0<x<1 \end{gathered}$$