# Download PDF by Daniel J. Velleman: American Mathematical Monthly, volume 116, number 1, january

By Daniel J. Velleman

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His favorite Hawaiian musician is Israel Kamakawiwo’ole. il 44 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 116 A Probabilistic Proof of the Lindeberg-Feller Central Limit Theorem Larry Goldstein 1. INTRODUCTION. The Central Limit Theorem, one of the most striking and useful results in probability and statistics, explains why the normal distribution appears in areas as diverse as gambling, measurement error, sampling, and statistical mechanics. In essence, the Central Limit Theorem states that the normal distribution applies whenever one is approximating probabilities for a quantity which is a sum of many independent contributions all of which are roughly the same size.

For even moderate values of n managing the binomial coefficients nk becomes unwieldy, to say nothing of computing the sum which yields the cumulative probability P(Sn ≤ m) = k≤m n k p (1 − p)n−k k that there will be m or fewer successes. January 2009] THE LINDEBERG-FELLER CENTRAL LIMIT THEOREM 45 The great utility of the CLT is in providing an easily computable approximation to such probabilities that can be quite accurate even for moderate values of n. To state this result, let Z denote a standard normal variable, that is, one with distribution function P(Z ≤ x) = (x) given by x (x) = −∞ ϕ(u)du where 1 1 ϕ(u) = √ exp − u 2 , 2 2π (2) and recall that we say a sequence of random variables Yn is said to converge in distribution to Y , written Yn →d Y , if lim P(Yn ≤ x) = P(Y ≤ x) n→∞ for all continuity points x of P(Y ≤ x).

E. Hearnshaw and M. S. Paterson, Problems drive, Eureka 27 (1964) 6–8 and 39–40. html. 11. C. P. Jargodzki and F. Potter, Mad About Physics: Braintwisters, Paradoxes, and Curiosities, John Wiley, New York, 2001. 12. P. B. Johnson, Leaning tower of lire, Amer. J. Phys. 23 (1955) 240. 13. G. M. , Clarendon, Oxford, 1907. 14. M. Paterson and U. Zwick, Overhang, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06), Society for Industrial and Applied Mathematics, Philadelphia, 2006, 231–240.