MAT 521Spring Lecture 15-16

# MAT 521Spring Lecture 15-16 - Example Find the pdf of the...

This preview shows pages 1–7. Sign up to view the full content.

1 Lecture 15-16: Section 3.9 II

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2
3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 The Range : X ..., , X , X n 2 1 a random sample ) X ..., , X , min(X Y n 2 1 1 = and ) X ..., , X , max(X Y n 2 1 n = W = Range of the sample = n Y - 1 Y What is the pdf of W? Define Z = 1 Y To derive the joint pdf of W and Z, we first derive the joint pdf of n Y and 1 Y : ) y Y and y Pr(Y - ) y Pr(Y ) y Y , y Y Pr( ) y , y ( G 1 1 n n n n n n 1 1 n 1 = = = ) y X y ..., , y X y , y X Pr(y - ) y Pr(Y n n 1 n 2 1 n 1 1 n n < < <
5 = < < = n 1 i n 1 1 n n ) y X y Pr( - ) y ( G = n 1 n n n )] y ( F ) [F(y - )] y ( F [ - The bivariate joint pdf of 1 Y and n Y can be found from the relation n 1 n 1 2 n 1 y y ) y , y ( G ) y , y ( g = . Thus for < < < - y y n 1 , ) y ( f ) y ( f )] F(y ) 1)[F(y - n(n ) y , y ( g n 1 2 n 1 n n 1 - - = . The transformation from 1 Y and n Y to W and Z is 1-1 linear transformation. The inverse transformation is specified by the following equations: 1 Y = Z, n Y = W + Z.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 The Jacobian |J| = 1. Therefore the joint pdf of W and Z is ) z w ( f ) z ( f )] F(z ) z 1)[F(w - n(n ) z , w ( h 2 n + - + = - . The marginal pdf of W can be obtained by taking the intergral of the above joint pdf.
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Example: Find the pdf of the range of a random sample from a uniform distribution. Assume that } X ..., , X , {X n 2 1 is a random sample from uniform [0, 1]. Then the pdf and cdf of any X is, respectively otherwise 1 x for 1 ) x ( f < < = and 1. x for 1 1 x for x x for ) x ( F ≥ < < ≤ = The joint pdf of W and Z is otherwise w-1 z and 1 w for 1)w-n(n ) z , w ( h 2-n < < < < = . 7 The pdf of W is ∫ < <--= =-w-1 o 1 n 2-n 1 otherwise 1 w for ) w 1 ( w ) 1 n ( n dz 1)w-n(n ) w ( h Practice Problems: #8, 9....
View Full Document

{[ snackBarMessage ]}

### Page1 / 7

MAT 521Spring Lecture 15-16 - Example Find the pdf of the...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online