lecture20 - Stat 302, Introduction to Probability Jiahua...

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Unformatted text preview: Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 20 January-April 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 January-April 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 January-April 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 January-April 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 !...
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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.

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lecture20 - Stat 302, Introduction to Probability Jiahua...

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