Q.6.30 - A First Course in Probability

Question

Compute the density of the range of a sample of size $n$ from a continuous distribution having density function $f$.

Step-by-Step Solution

Answer

Density of a sample of size $n$ is $P(R\le a)=n{\int }_{-}^{\infty }{\left[F(x_1+a)-F\left(x_1\right)\right]}^{-1}f\left(x_1\right)d{x}_1$.

Step 1 : Joint probability distribution :

The related probability distribution on all possible pairings of outputs is the joint probability distribution. For each given number of random variables, the joint distribution may be studied.

Step 2 : Explanation :

Density function $f$ and sample size $n$,

$$\begin{gathered} P\left\{R\le a\right\}=P\left\{X_{\left(n\right)}-X_{\left(0\right)}\le a\right\} \\ ={\iint }_{x_{\left(i\right)}-x_{\left(i\right)}} {\int }_{x_i}{x}_n\left(x_1{x}_n\right)d{x}_1......d{x}_n \\ ={\int }_{-x}^{\infty }{\int }_{-\infty }^{x_0+a}\frac{n!}{(n-2)!}{\left[F\left(x_n\right)-F\left(x_1\right)\right]}^{n-2}f\left(x_1\right)f\left(x_n\right)d{x}_{1}d{x}_n \end{gathered}$$

Putting $y=F\left(x_n\right)-F\left(x_1\right), dy=f\left(x_n\right)d{x}_n$

We get,

$$\begin{gathered} {\int }_{-x}^{\infty }{\int }_x^{y+a}{\left[F\left(x_n\right)-F\left(x_1\right)\right]}^{n-2}f\left(x_n\right)d{x}_n={\int }_{-\infty }^{\infty }{\int }_0^{F(x_1+a)-F\left(x_1\right)}{y}^{n-2}dy \\ ={\int }_{-\infty }^{\infty }\frac{1}{n-1}{\left[F(x_1+a)-F\left(x_1\right)\right]}^{n-1} \end{gathered}$$

Thus, $P(R\le a)=n{\int }_{-}^{\infty }{\left[F\left(x_1+a\right)-F\left(x_1\right)\right]}^{-1}f\left(x_1\right)d{x}_1$.