مبرهنة النهاية المركزية

تشكل مبرهنات النهاية المركزية مجموعة نتائج لنظرية الإحتمالات تنص أن مجموع عدة متغيرات عشوائية مستقلة و متشابهة التوزع ، يميل إلى التوزع حسب توزيع احتمالي معين .

Histogram plot of average proportion of heads in a fair coin toss, over a large number of sequences of coin tosses.

أهم هذه المبرهنات تقول أنه اذا كانت المتغيرات المجموعة تملك تباينات محددة فإن المجموع يميل إلى التوزع طبيعيا أي أنه يملك توزيعا احتماليا طبيعيا .

تسمى مبرهنة النهاية المركزية أيضاً، وبالمبرهنة الأساسية الثانية في الإحصاء.

لتكن X1, X2, X3, ... Xn متسلسله من الاعدادالمستقله والمتطابقه في التوزيع

المتغير العشوائي لكل منها لديه قيمه منتهي للوسط µ و التباين σ2 > 0.

تقول مبرهنة النهايه المركزيه ان:كلما ازداد حجم العينه n ,فان التوزيع لمتوسط هذه المتغيرات العشوائيه يقترب من التوزيع الطبيعي .

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 مبرهنة النهاية المركزية الكلاسيكية

Pictures of a distribution being "smoothed out" by summation (showing original density of distribution and three subsequent summations). (See Illustration of the central limit theorem for further details.)

 تطبيقات وأمثلة

A histogram plot of monthly accidental deaths in the US, between 1973 and 1978 exhibits normality, due to the central limit theorem.

There are a number of useful and interesting examples and applications arising from the central limit theorem (Dinov, Christou & Sanchez 2008). See e.g. [1], presented as part of the SOCR CLT Activity.

• The probability distribution for total distance covered in a random walk (biased or unbiased) will tend toward a normal distribution.
• Flipping a large number of coins will result in a normal distribution for the total number of heads (or equivalently total number of tails).

 انظر ايضاً

• Diversification (finance)
• Illustration of the central limit theorem
• Law of large numbers — weaker conclusion in the same context
• Log-normal distribution‎ — what we get when we multiply random variables in a similar context to the Central limit theorem
• Berry–Esséen theorem –— error bounds on normal approximations based on the central limit theorem

 المصادر

• Henk Tijms, Understanding Probability: Chance Rules in Everyday Life, Cambridge: Cambridge University Press, 2004.
• S. Artstein, K. Ball, F. Barthe and A. Naor (2004), "Solution of Shannon's Problem on the Monotonicity of Entropy", Journal of the American Mathematical Society 17, 975-982. Also author's site.
• S.N.Bernstein, On the work of P.L.Chebyshev in Probability Theory, Nauchnoe Nasledie P.L.Chebysheva. Vypusk Pervyi: Matematika. (Russian) [The Scientific Legacy of P. L. Chebyshev. First Part: Mathematics] Edited by S. N. Bernstein.] Academiya Nauk SSSR, Moscow-Leningrad, 1945. 174 pp.
• G. Rempala and J. Wesolowski, "Asymptotics of products of sums and U-statistics", Electronic Communications in Probability, vol. 7, pp. 47-54, 2002.
• Dinov, Ivo; Christou, Nicolas; Sanchez, Juana (2008), "Central Limit Theorem: New SOCR Applet and Demonstration Activity", Journal of Statistics Education (ASA) 16 (2) . Also at ASA/JSE.
• Billingsley, Patrick (1995), Probability and Measure (Third ed.), John Wiley & sons, ISBN 0-471-00710-2 
• Rice, John (1995), Mathematical Statistics and Data Analysis (Second ed.), Duxbury Press, ISBN 0-534-20934-3 
• Durrett, Richard (1996), Probability: theory and examples (Second ed.) 
• Klartag, Bo'az (2007), "A central limit theorem for convex sets", Inventiones Mathematicae 168, 91-131. Also arXiv.
• Klartag, Bo'az (2008), "A Berry-Esseen type inequality for convex bodies with an unconditional basis", Probability Theory and Related Fields. Also arXiv.
• Le Cam, Lucien (1986), "The central limit theorem around 1935", Statistical Science 1:1, 78-91.
• Zygmund, Antoni (1959), Trigonometric series, II, Cambridge .
• Barany, Imre; Vu, Van (2007), "Central limit theorems for Gaussian polytopes", The Annals of Probability (Institute of Mathematical Statistics) 35 (4): 1593-1621, doi:10.1214/009117906000000791 . Also arXiv.
• Meckes, Elizabeth (2008), "Linear functions on the classical matrix groups", Transactions of the American Mathematical Society 360: 5355-5366, doi:10.1090/S0002-9947-08-04444-9 . Also arXiv.
• Gaposhkin, V.F. (1966), "Lacunary series and independent functions", Russian Math. Surveys 21 (6): 1-82, doi:10.1070/RM1966v021n06ABEH001196 .

 وصلات خارجية

• Animated examples of the CLT
• Central Limit Theorem Java
• Central Limit Theorem interactive simulation to experiment with various parameters
• CLT in NetLogo (Connected Probability - ProbLab) interactive simulation w/ a variety of modifiable parameters
• General Central Limit Theorem Activity & corresponding SOCR CLT Applet (Select the Sampling Distribution CLT Experiment from the drop-down list of SOCR Experiments)
• Generate sampling distributions in Excel Specify arbitrary population, sample size, and sample statistic.
• [2] Another proof.
• CAUSEweb.org is a site with many resources for teaching statistics including the Central Limit Theorem
• The Central Limit Theorem by Chris Boucher, Wolfram Demonstrations Project.
• Eric W. Weisstein, Central Limit Theorem at MathWorld.
• Animations for the Central Limit Theorem by Yihui Xie using the R package animation
• A visualization of the Central Limit Theorem from Portfolio Monkey.