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Binomial Approximations

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DeMoivre-Laplace Equation

If ~X is binomially distributed random variable with number parameter ~n, and probability parameter &theta.

~X _ ~ _ B ( ~n , &theta. ) , _ _ p ( ~x ) _ = _ b ( ~x ; ~n , &theta. ) _ = _ comb{~n,~x} &theta.^~x ( 1 - &theta. )^{~n - ~x}

~X _ has mean ~n &theta. , and variance ~n &theta. ( 1 - &theta. ).

If ~n is very large, ~X is approximately normally distributed, with the same mean and variance, i.e.

p ( ~y =< ~x ) _ ~~ _ &Phi.rndb{fract{~x - ~n &theta.,&sqrt.${~n &theta. ( 1 - &theta. )}}}

where &Phi. is the distribution function of the standard normal distribution .

 

 

Maximum Likelihood Estimators