This preview shows pages 1–10. 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 DocumentThis 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 DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Statistics 150: Spring 2007 January 22, 2008 01 NOTE: These slides are not meant to be complete lecture notes that replace attending class. Rather, they are intended to act as a basis for the discussion in class and you will need to attend class to flesh out the details. 1 1 Examples of discrete variables Bernoulli trials. A Bernoulli random variable X takes values 1 and 0 with probability p and q (= 1 p ) , respectively. Sometimes we think of these values as representing the success or the failure of a trial. The mass function is f (0) = 1 p, f (1) = p, and it follows that E [ X ] = p and var [ X ] = p (1 p ) . 2 Binomial distribution. Perform n independent, identically distributed Bernoulli trials X 1 ,X 2 ,...,X n and count the total number of successes Y = X 1 + X 2 + + X n . The mass function of Y is f ( k ) = n k p k (1 p ) n k , k = 0 , 1 ,...,n. and we can compute that E [ Y ] = np, var [ Y ] = np (1 p ) . 3 Poisson distribution. A Poisson variable is a random variable with the Poisson mass function f ( k ) = k k ! e , k = 0 , 1 , 2 ,... for some . Suppose Y be a bin( n,p ) variable and let n and p together in such a way that E [ Y ] = np approaches a nonzero constant . Then, for k = 0 , 1 , 2 ,..., P { Y = k } = n k p k (1 p ) n k 1 k ! np 1 p k (1 p ) n k k ! e . * Check that both the mean and the variance of this distribution are equal to . 4 2 Total variation distance Let F and G be the distribution functions of discrete distributions which place masses f n and g n at the points x n , for n 1 , and define d TV ( F,G ) = X k 1  f k g k  . The quantity d TV ( F,G ) is called the total variation distance between F and G . 5 For random variables X and Y , define d TV ( X,Y ) = d TV ( F X ,F Y ) . Note that d TV ( X,Y ) = 2 sup A R  P { X A }  P { Y A } for discrete random variables X,Y . 6 3 Poisson convergence Theorem 3.1. Let { X r : 1 r n } be independent Bernoulli random variables with respective parameters { p r : 1 r n } , and let S = n r =1 X r . Then d TV ( S,P ) 2 n X r =1 p 2 r where P is a random variable having the Poisson distribution with parameter = n r =1 p r . 7 Proof. Let ( X r ,Y r ) , 1 r n , be a sequence of independent pairs, where the pair ( X r ,Y r ) takes values in the set { , 1 } { , 1 , 2 ,... } with mass function P { X r = x,Y r = y } = 1 p r , if x = y = 0; e p r 1 + p r , if x = 1 ,y = 0; p y r y ! e p r , if x = 1 ,y 1 . It is easy to check that X r is Bernoulli with parameter p r , and Y r has the Poisson distribution with parameter p r ....
View Full
Document
 Spring '08
 Evans
 Statistics

Click to edit the document details