08_26 - STAT 410 Examples for 08/26/2011 Transformations of...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 410 Examples for 08/26/2011 Fall 2011 Transformations of Random Variables Example 1: x p X ( x ) Y = X 2 y = x 2 p Y ( y ) = p X ( y ) 1 0.2 1 0.2 2 0.4 4 0.4 3 0.3 9 0.3 4 0.1 16 0.1 Example 2: x p X ( x ) Y = X 2 y p Y ( y ) 2 0.2 0 p X ( 0 ) = 0.4 0 0.4 4 p X ( 2 ) + p X ( 2 ) = 0.5 2 0.3 9 p X ( 3 ) = 0.1 3 0.1 Example 3: X ~ Poisson ( λ ): p X ( x ) = ! λ λ x e x - , x = 0, 1, 2, 3, 4, … . Y = X 2 p Y ( y ) = ( ) ! λ λ y e y - , y = 0, 1, 4, 9, 16, … .
Background image of page 1

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

View Full DocumentRight Arrow Icon
Let X be a continuous random variable. Let Y = g ( X ). What is the probability distribution of Y ? Cumulative Distribution Function approach: F Y ( y ) = P ( Y y ) = P ( g ( X ) y ) = ( ) ( ) { } : X y x x g dx x f = Moment-Generating Function approach: M Y ( t ) = E ( e Y t ) = E ( e g ( X ) t ) = ( ) ( ) - dx x f t x g e X = 1. Let U be a Uniform ( 0, 1 ) random variable: f U ( u ) = < < o.w. 0 1 0 1 x F U ( u ) = < < 1 1 1 0 0 0 u u u u Consider Y = U 2 . What is the probability distribution of Y ? F Y ( y ) = P ( Y y ) = P ( U 2 y ) y < 0 P ( U 2 y ) = 0 F Y ( y ) = 0. 0 y < 1 P ( U 2 y ) = P ( U y ) = y F Y ( y ) = y . y 1 P ( U 2 y ) = 1 F Y ( y ) = 1. f Y ( y ) = F ' Y ( y ) = ( ) < < otherwise 0 1 0 2 1 y y
Background image of page 2
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Theorem 1.7.1 X continuous r.v. with p.d.f. f X ( x ). Y = g ( X ) g ( x ) one-to-one, differentiable d x / d y = d [ g 1 ( y ) ] / d y f Y ( y ) = f X ( g 1 ( y ) ) y x d d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ g ( u ) = u 2 g 1 ( y ) = y = y 1 / 2 d u / d y = 2 1 y 1 / 2 f Y ( y ) = f U ( g 1 ( y ) ) y u d d = ( 1 ) | 2 1 y 1 / 2 | = 2 1 y 1 / 2 0 < y < 1 2. Consider a continuous random variable X with p.d.f. f X ( x ) = < < o.w. 0 1 0 2 x x a) Find the probability distribution of Y = X . f
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

Page1 / 14

08_26 - STAT 410 Examples for 08/26/2011 Transformations of...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online