16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde 3. The Poisson approximation to the binomial distribution The binomial distribution, like the Poisson, is that of a random variable taking only positive integral values. Since it involves factorials, the binomial distribution is not very convenient for numerical application We shall show under what conditions the poisson expression serves as a good approximation to the binomial expression-and thus may be used for g00 convenience b(k)= p(1-p k!(n-k Consider a large number of trials, n, with small probability of success in each, p, such that the mean of the distribution, np, is of moderate magnitude. Define u= np with n large and p small P Recalling 2n 2en Stirlings formula b(k)= k!(n-k)! n 2丌n2e A2T(n-k)2e-+k n 2e k! as n becomes large relative to k ue The relative error in this approximation is of order of magnitude Rel. Error≈ 9/30/2004955AM Page 2 of 1016.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde 9/30/2004 9:55 AM Page 2 of 10 3. The Poisson Approximation to the Binomial Distribution The binomial distribution, like the Poisson, is that of a random variable taking only positive integral values. Since it involves factorials, the binomial distribution is not very convenient for numerical application. We shall show under what conditions the Poisson expression serves as a good approximation to the binomial expression – and thus may be used for convenience. ( ) ! ( ) (1 ) ! ! n k nk bk p p knk − = − − Consider a large number of trials, n, with small probability of success in each, p, such that the mean of the distribution, np, is of moderate magnitude. 1 2 1 2 1 2 1 2 1 2 Define with large and small Recalling: ! ~ 2 Stirling's formula lim 1 ! () 1 !( )! 2 1 ! 2( ) ! 1 n n n n n k k k n n k k n n nk k n k n n np n p p n n ne e n n b k kn k n n n e k n nk e n n e k n µ µ µ π µ µ µ µπ µ π µ + − − →∞ − + − − + − + + − + ≡ = ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = − ⎜ ⎟ − ⎝ ⎠ ⎛ ⎞ ≈ − ⎜ ⎟ ⎝ ⎠ − = 1 2 1 as becomes large relative to ! 1 ! n k n k k k k k k n k e n e n k kee e k µ µ µ µ µ − − + − − − ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ − ⎝ ⎠ ≈ = The relative error in this approximation is of order of magnitude 2 ( ) Rel. Error ~ k n − µ