This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: week 7 1 Functions of Random variables In some case we would like to find the distribution of Y = h ( X ) when the distribution of X is known. Discrete case E x a m p l e s 1. Let Y = aX + b , a 2. Let ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = = y h x Y x X P y h X P y X h P y Y P y p 1 1 ( ) ( ) ( ) = = = + = = b y a X P y b aX P y Y P 1 2 X Y = ( ) ( ) ( ) ( ) ( ) < = = > = + = = = = = 2 y if y if X P y if y X P y X P y X P y Y P week 7 2 Continuous case Examples 1. Suppose X ~ Uniform(0, 1). Let , then the cdf of Y can be found as follows The density of Y is then given by 2. Let X have the exponential distribution with parameter . Find the density for 3. Suppose X is a random variable with density Check if this is a valid density and find the density of . 2 X Y = 1 1 + = X Y ( ) + = elsewhere x x x f X , 1 1 , 2 1 ( ) ( ) ( ) ( ) ( ) y F y X P y X P y Y P y F X Y = = = = 2 2 X Y = week 7 3 Question Can we formulate a general rule for densities so that we dont have to look...
View
Full
Document
 Fall '10
 MCDUNNOUGH
 Probability

Click to edit the document details