Example_class_8_handout_solution0

# Example_class_8_handout_solution0 - THE UNIVERSITY OF HONG...

This preview shows pages 1–3. Sign up to view the full content.

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 8 Review Joint and Marginal Distributions 1. Let X 1 ,...,X n be random variables deﬁned on the same sample space Ω . The joint distribution function of ( X 1 ,...,X n ) is deﬁned by F ( x 1 ,...,x n ) = P ( X 1 x 1 ,X 2 x 2 ,...,X n x n , ) The distribution function F X i of each X i is called the marginal distribution function of X i . 2. For discrete random variables X 1 ,...,X n , the joint probability mass function of ( X 1 ,...,X n ) is f ( x 1 ,...,x n ) = P ( X 1 = x 1 ,X 2 = x 2 ,...,X n = x n , ) The mass function f X i = P ( X i = x ) of each X i is called the marginal probability mass function of X i . f X i ( x ) = X u 1 ... X u i - 1 X u i +1 ... X u n f ( u 1 ,...,u i - 1 ,x,u i +1 ,...,u n ) 3. Random variables X 1 ,...,X n are (jointly) continuous if their joint distribution function F satisﬁes F ( x 1 ,...,x n ) = ˆ x 1 -∞ ... ˆ x n -∞ f ( u 1 ,...,u n ) du n ...du 1 , for some nonnegative function f : ( -∞ , ) n [0 , ) . The function f is called the joint probability density function of ( X 1 ,...,X n ) . The pdf of X i is called the marginal pdf of X i . f X i ( x ) = ˆ -∞ ... ˆ -∞ f ( u 1 ,...,u i - 1 ,x,u i +1 ,...,u n ) du n ...du i +1 du i - 1 ...du 1 . 4. If a joint distribution function F possesses all partial derivatives at ( X 1 ,...,X n ) , then the joint pdf is f ( x 1 ,...,x n ) = n ∂x 1 ...∂x n F ( x 1 ,...,x n ) . Independence of random variables Random variables X 1 ,...,X n are independent if and only if their joint pmf(pdf) or cdf is equal to the product of their marginal pmfs(pdfs) or cdfs, i.e. f ( x 1 ,...,x n ) = f X 1 ( x 1 ) ...f X n ( x n ) or F ( x 1 ,...,x n ) = F X 1 ( x 1 ) ...F X n ( x n ) Proposition Random variables X and Y are independent if and only if 1. the supports of X and Y do not depend on each other 2. f ( x,y ) can be factorized as g ( x ) h ( y ) This proposition applies to both discrete and continuous random variables and can be generalized to multivariate cases. 1

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

View Full Document
Expectation of function of random variables Deﬁnition For random variables X 1 ,...,X n with joint pmf or pdf f ( x 1 ,...,x n ) , if u ( X 1 ,...,X n ) is a function of these random variables, then the expectation of this function is deﬁned as E ( u ( X 1 ,...,X
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

### Page1 / 7

Example_class_8_handout_solution0 - THE UNIVERSITY OF HONG...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online