Question

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

Short 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-by-step solution

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.

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