مبرهنة النهاية المركزية

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

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 .

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 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 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.