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In this lecture, we describe two proofs of a central theorem of mathemat-ics, namely the central limit theorem. One will be using cumulants, and the other using moments. Actually, our proofs won’t be entirely formal, but we will explain how to make them formal.
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In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed.
Proof of the Central Limit Theorem Theorem: Let X1;X2;:::;Xn be a random sample of size n from N( ;˙2). De ne X = 1 n Pn i=1 Xi, then ˘ N( ;˙2=n). Theorem: Let X be the mean of a random sample X1;X2;:::;Xn of size n from a distri-bution with mean and variance ˙2 n
The central limit theorem is a theorem about independent random variables, which says roughly that the probability distribution of the average of independent random variables will converge to a normal distribution, as the number of observations increases.
The sum of i.i.d. random variables is normally distributed with mean and variance %. Proof: The Fourier Transform of a PDF is called a characteristic function. Take the characteristic function of the probability mass of the sample distance from the mean, divided by standard deviation.
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棣莫佛-拉普拉斯定理（De Moivre–Laplace theorem）是中央极限定理的最初版本，讨论了服从二项分布的随机变量序列。它指出，参数为n, p的二项分布以np为均值、np(1-p) 为方差的正态分布为极限。
n!1. Sn np. p. npq. bg ! (b) (a): Let Xi be an i.i.d. sequence of random variables. Write Sn = Pn i=1 Xn. Suppose each Xi is 1 with probability p and 0 with probability. q = 1 p. DeMoivre-Laplace limit theorem: lim Pfa. n!1. Sn np. p. npq. bg ! (b) (a): Here (b) (a) = Pfa normal random variable.
