MAT 521Spring Lecture 15-16

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

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

View Full Document Right Arrow Icon
1 Lecture 15-16: Section 3.9 II
Background image of page 1

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

View Full DocumentRight Arrow Icon
2
Background image of page 2
3
Background image of page 3

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

View Full DocumentRight Arrow Icon
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 < < <
Background image of page 4
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.
Background image of page 5

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

View Full DocumentRight Arrow Icon
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.
Background image of page 6
Background image of page 7
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 &lt; &lt; = and 1. x for 1 1 x for x x for ) x ( F &lt; &lt; = 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 &lt; &lt; &lt; &lt; = . 7 The pdf of W is &lt; &lt;--= =-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

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 Right Arrow Icon
Ask a homework question - tutors are online