KATTA SONLAR QONUNI VA MARKAZIY LIMIT TEOREMASINING STATISTIK TAHLILDAGI AHAMIYATI
Keywords:
Kalit so‘zlar: katta sonlar qonuni, markaziy limit teoremasi, matematik kutilma, dispersiya, tanlanma o‘rtachasi, Chebishev tengsizligi, Bernulli sxemasi, normal taqsimot, statistik xulosalashAbstract
ANNOTATSIYA
Ushbu maqolada katta sonlar qonuni va markaziy limit teoremasining statistik
tahlildagi nazariy hamda amaliy ahamiyati batafsil ko‘rib chiqildi. Tadqiqotda
tasodifiy miqdor, matematik kutilma, dispersiya, Chebishev tengsizligi, Bernulli
sxemasi va normal taqsimotning o‘zaro bog‘liqligi tizimli ravishda yoritildi. Katta
sonlar qonuni tanga tashlash tajribasi orqali tekshirildi: 100, 500, 1000 va 5000
martalik sinovlar uchun nisbiy chastotalar hisoblanib, ular nazariy ehtimollik bilan
solishtirildi. Markaziy limit teoremasi esa bir xil taqsimlangan tasodifiy sonlardan
olingan tanlanma o‘rtachalarining histogrammasi va ularning standart og‘ishining
kamayish sur’ati orqali namoyish etildi. Natijalar kuzatuvlar soni oshgani sari empirik
baholarning barqarorlashishini, o‘rtachalar taqsimotining normal qonunga
yaqinlashishini va statistik xulosalashning ishonchliligi kuchayishini ko‘rsatdi. Maqola
yakunida iqtisodiyot, sug‘urta, bank va ma’lumotlar tahlili kabi sohalarda ushbu
teoremalar qanday amaliy rol o‘ynashi muhokama qilinadi.
References
ADABIYOTLAR
Borsato, L., Horta, E., & Souza, R. R. (2024). Product disintegrations: A law of large
numbers via conditional independence. Statistics & Probability Letters, 208, 110056.
https://doi.org/10.1016/j.spl.2024.110056
Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
Durrett, R. (2019). Probability: Theory and examples (5th ed.). Cambridge University
Press.
Horvath, B., Jacquier, A., Muguruza, A., & Søjmark, A. (2024). Functional central
limit theorems for rough volatility. Finance and Stochastics, 28, 615–661.
https://doi.org/10.1007/s00780-024-00533-5
Kwak, S. G., & Kim, J. H. (2017). Central limit theorem: The cornerstone of modern
statistics. Korean Journal of Anesthesiology, 70(2), 144–156.
https://doi.org/10.4097/kjae.2017.70.2.144
Liang, K., & Zhang, Y. (2024). Almost sure central limit theorem for error variance
estimator in pth-order nonlinear autoregressive processes. Mathematics, 12(10), 1482.
https://doi.org/10.3390/math12101482
Ross, S. M. (2019). A first course in probability (10th ed.). Pearson.
Shanafelt, D. W. (2025). How much is enough? Applying the law of large numbers to
the measurement of interactions between ecosystem services. Ecosystem Services, 74,
101736. https://doi.org/10.1016/j.ecoser.2025.101736
Ulyanov, V. V. (2024). From classical to modern nonlinear central limit theorems.
Mathematics, 12(14), 2276. https://doi.org/10.3390/math12142276
Vogel, R. M., Papalexiou, S. M., Lamontagne, J. R., & Dolan, F. C. (2025). When
heavy tails disrupt statistical inference. The American Statistician, 79(2), 221–235.
