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

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Stat 302, Introduction to Probability Jiahua Chen January-April 2011 Jiahua Chen () Lecture 20 January-April 2011 1 / 16

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Convergence in distribution Let X n be a binomial random variable with parameter n and p = μ / n . It is seen that when n , p 0, while np = μ 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 = μ / n . Note that P ( X n = 5 ) = n ! 5 ! ( n - 5 ) ! parenleftBig μ n parenrightBig 5 parenleftBig 1 - μ n parenrightBig n - 5 . It is easy to see that as n . parenleftBig 1 - μ n parenrightBig n - 5 exp ( - μ ) Jiahua Chen () Lecture 20 January-April 2011 3 / 16

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Poisson approximation to Binomial Let X n be a binomial random variable with parameter n and p = μ / n . Note that P ( X n = 5 ) = n ! 5 ! ( n - 5 ) ! parenleftBig μ n parenrightBig 5 parenleftBig 1 - μ n parenrightBig n - 5 . It is easy to see that as n . parenleftBig 1 - μ n parenrightBig n - 5 exp ( - μ ) Does the factor n ! 5 ! ( n - 5 ) ! parenleftBig μ 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 μ n parenrightBig 5 = n ( n - 1 )( n - 2 )( n - 3 )( n - 4 ) n * n * n * n * n × μ 5 5 ! μ 5 5 ! . Combining two limits, we find P ( X n = 5 ) = n ! 5 ! ( n - 5 ) ! parenleftBig μ n parenrightBig 5 parenleftBig 1 - μ n parenrightBig n - 5 μ 5 5 !

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lecture20 - Stat 302 Introduction to Probability Jiahua...

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