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《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:

2014年03期

Page:

39-

Research Field:

数学

Publishing date:

Info

Title:

Asymptotic Expansions of the Probability Density Function and theDistribution Function of Chi-Square Distribution

Author(s):

Chen Gang¹; Wang Mengjie²

(1.Basic Course Department,Nantong Vocation University,Nantong 226007 China)(2.School of Business,Centennial College,Toronto M1K 5E9,Canada)

Keywords:

χ² distribution; probability density function; distribution function; asymptotic expansion; standard transformation

PACS:

O211

DOI:

-

Abstract:

Through the transformation of the independent variable of χ² distribution probability density function,degree of freedom of which is n,the equation can be expanded as follows: (2n)^1/2χ²(x; n)=f(t; n)=[1+(r₁(t))/(n^1/2)+(r₂(t))/n+(r₃(t))/(nn^1/2)+(r₄(t))/(n²)] φ(t)+o(1/(n²)),here,φ(t)is a density function of standard normal distribution; r_i(t)is a 3i order polynomial of t(1≤i≤4).An approximate formula can be obtained from the expansion of the distribution density function.We further establish the integral recurrence relations of the power coefficients of the standard normal density function and obtain the asymptotic expansion of the distribution function of χ².Finally,the effectiveness of these results in practical application was verified by the numerical calculations.

References:

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Memo

Memo:

-

Common functions

Last Update: 2014-09-30