The Application Of Approximation Methods For Normal Distribution

In real life, we are accustomed to analyze problems from thestatistical aspect, which needs us to consider things in our lives as randomvariables. To further investigate these random variables, we introduce thedistribution functions, which provides lots of assistance to the progress ofresearching the random variables. However, even though we already knowthe distributions of these random variables, we can not determine theparameters of the distribution functions. For this purpose, madow(1948),erdos and renyi(1959) , hajek(1960) [3]introduced central limittheorem(CLT) which proved many kinds of distribution could beestimated by normal distribution. This paper is based on the CLT andsummarized the functions that can be simulated by normal distributionand then, discuss the errors.Hypergeometric distribution has a wide range of applications,including statistical quality control , capture–recapture methods , samplesurveys ,analysis of contingency tables , etc. When the number of simplesis big enough, the parameters of Hypergeometric distribution are difficultto determine. Thus, what we concern is whether we can use normaldistribution to estimate Hypergeometric distribution. After a mass of reading, it is found that the approximation with respect to hypergeometricdistribution by using normal distribution is based on the way of applyingnormal distribution to approach binomial distribution. More interesting,the limit of hypergeometric distribution is binomial distribution.Therefore, this paper summarized the developing process of using normaldistribution to simulate hypergeometric distribution.First of all, I introduced the relationship between binomialdistribution and normal distribution, i.e., the differentiation function ofnormal distribution is derived from binomial distribution, which gives anidea that using normal distribution to approach binomial distribution[34].Hence, this paper introduced the DeMoire-Laplace Limit Theorem[33] .DeMoire-LaplacDeMoire-Laplace with , and is the binomial distribu- tion function.In other words, the relative error of the above formula tends to zero, with1 3 , 1 3 tend to zero. Using S.Berstein Theorem, Feller(1945)[14] v z z ? ? ? ? ?improved the above results to the more accurately approximate the results.On the basis of Feller's result, Nicholson(1956)[6] studiedhypergeometric distribution similarly and develop a method of using normal distribution to simulate hypergeometric distribution . But, Lahiri,Chatterjee and Maiti(2007)[17] found that there was a great deviationbetween normal distribution and hypergeometric distribution in the case ofnon-standard. In order to describe the devation, they gave an improvedversion of Berry-Esseen Theorem [17]. And the improved theorem includeda criterion to determine whether normal distribution could simulatehypergeometric distribution. At last, I introduce the criterion given byLahiri and Chatterjee(2007)[15].(1)Binomial distribution[14]:In case of using normal distribution to simulate , when , Feller pointed out that can beestimated byWhen applying the second , the inequality behind is converse. k冐,v P冘Then the estimation is about the first interval of . ,v P冘2hypergeometric distribution: In case of using normal distribution to simulate , it is very ,complicated:If p and s approach to 0.5 (with 1 , 1 ) , i.e. thedistribution of the sample is symmetry of the mean value, then that can beestimated bywith 1 2 ( ) ;If the distribution of the sample is not symmetry, then we can use themethod given by Nicholson. First, making a judement that whether. Then on basis of in-equalitywe can compute the interval of the real value of , and give the point ,v H ?estimation of . Denoted the point estimation and the upper bound and ,v H ?lower bound by , then the upper bound of the corresponding ^Let the above value be the normal distribution estimation of . And let ,v H ? However,after studying, Lahiri, Chatterjee and Maiti found that theaccuracy of the Normal approximation to the Hypergeometric distributiondeteriorates as p=D/N or f=n/N tend to the boundary value 1.In order todescribe the deviation, Lahiri, Chatterjee and Maiti gives improved Berry-Esseen Theorem. If> 1, we can use the following inequationTo determine the error in the process of using normal distribution tosimulate hypergeometric distribution, then ? ? 0 with,1 2 C ,C ?(0,?) In non-standard case, there is a great deviation in the process of using normal distribution to simulate hypergeometricdistribution. So Lahiri, Chatterjee and Maiti develop a criterion todetermine whether normal distribution could simulate hypergeometricdistribution.Defined coefficient , and satisfy that For the given (N , p, f ) , compute the value of ? (N , p, f ) and look up thevalue of c (N , p , f ) from the Table3.2~3.5in§3.3. Then compute?(N ,p, f ) from the above equation and compare with the one computedfrom the following equation .If the difference is not large,we consider that non-standardhypergeometric distribution can be estimated by normal distribution；andthe result will be reverse if it is on the contrary.In addition,Lahiri and Chatterjee derive the conditions that can beused as normal distribution approximates to hypergeometric distribution.If 1 , and . Then with , and normal variable , we consider that hypergeometric distribution can be estimatedby normal distribution .