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

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

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, … .

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

View Full Document
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
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 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

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

View Full Document
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
Ask a homework question - tutors are online