6 Example Class 5

6 Example Class 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

This preview shows pages 1–2. Sign up to view the full content.

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.

Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 5 Review 1. Expectation Assume that E ( X ) and E ( g ( X )) exist in the following, then E ( X ) E ( g ( X )) X is a discrete r.v. ) ( ) ( X x x xp ) ( ) ( ) ( X x x p x g X is a continuous r.v. ) ( ) ( X dx x xf ) ( ) ( ) ( X dx x f x g Mean and variance Mean of X = E ( X ) Variance of X = Var ( X ) = ( ) 2 )) ( ( X E X E- = E ( X 2 ) E ( X ) 2 The mean is usually denoted by and the variance is usually denoted by 2 . 2. Moments and moment generating function Moment Let r be a positive integer. The r th moment of X is E ( X r ). The r th moment of X about b is ( ) r b X E ) (- . Moment generating function Let X be a random variable. The moment generating function of X is defined as M X ( t ) = E ( e tX ) if it exists. The domain of g G is all real number t such that the expectation above is finite. And we have g G u = E ( X r ) if M X ( t ) is differentiable r times at t = 0. Remark: moment generating function uniquely characterizes the distribution....
View Full Document

This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

Page1 / 4

6 Example Class 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online