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Chapter 6: Q.6.30 (page 277)

Compute the density of the range of a sample of size nfrom a continuous distribution having density function f.

Short Answer

Expert verified

Density of a sample of size nis P(R鈮�a)=n鈭�-鈭�F(x1+a)-F(x1)-1f(x1)dx1.

Step by step solution

01

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.

02

Explanation :

Density function fand sample size n,

PR鈮�a=PXn-X0鈮�a=鈭�xi-xi鈭�xixn(x1xn)dx1......dxn=鈭�-x鈭�鈭�-鈭�x0+an!(n-2)!F(xn)-F(x1)n-2f(x1)f(xn)dx1dxn

Putting y=F(xn)-F(x1),dy=f(xn)dxn

We get,

鈭�-x鈭�鈭�xy+aF(xn)-F(x1)n-2f(xn)dxn=鈭�-鈭�鈭�鈭�0F(x1+a)-F(x1)yn-2dy=鈭�-鈭�鈭�1n-1F(x1+a)-F(x1)n-1

Thus,P(R鈮�a)=n鈭�-鈭�Fx1+a-F(x1)-1f(x1)dx1.

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