This preview shows pages 1–6. 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 Document
Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen JanuaryApril 2011 Jiahua Chen () Lecture 20 JanuaryApril 2011 1 / 16 Convergence in distribution Let X n be a binomial random variable with parameter n and p = / n . It is seen that when n , p 0, while np = is stationary. We find that P ( X n = k ) has a limit for any k . Let us illustrate this case when k = 5. Jiahua Chen () Lecture 20 JanuaryApril 2011 2 / 16 Poisson approximation to Binomial Let X n be a binomial random variable with parameter n and p = / n . Note that P ( X n = 5 ) = n ! 5 ! ( n 5 ) ! parenleftBig n parenrightBig 5 parenleftBig 1 n parenrightBig n 5 . It is easy to see that as n . parenleftBig 1 n parenrightBig n 5 exp ( ) Jiahua Chen () Lecture 20 JanuaryApril 2011 3 / 16 Poisson approximation to Binomial Let X n be a binomial random variable with parameter n and p = / n . Note that P ( X n = 5 ) = n ! 5 ! ( n 5 ) ! parenleftBig n parenrightBig 5 parenleftBig 1 n parenrightBig n 5 . It is easy to see that as n . parenleftBig 1 n parenrightBig n 5 exp ( ) Does the factor n ! 5 ! ( n 5 ) ! parenleftBig n parenrightBig 5 also have a limit? Jiahua Chen () Lecture 20 JanuaryApril 2011 3 / 16 Limit of the other factor Let us work as follows: n ! 5 ! ( n 5 ) ! parenleftBig n parenrightBig 5 = n ( n 1 )( n 2 )( n 3 )( n 4 ) n * n * n * n * n 5 5 ! 5 5 ! . Combining two limits, we find P ( X n = 5 ) = n !...
View
Full
Document
This note was uploaded on 04/07/2011 for the course STAT 302 taught by Professor Dr.chen during the Spring '11 term at The University of British Columbia.
 Spring '11
 Dr.Chen
 Binomial, Probability

Click to edit the document details