Q.6.49 - A First Course in Probability

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 $X_{\left(1\right)=S_1},X_{\left(2\right)=S_2},....,X_{(n-1)=S(n-1)}$.

Step-by-Step Solution

Answer

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

Step 1 : 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.

Step 2 : 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.

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