§2. The central limit theorem

1. Convergence in distribution Suppose that {X n } are i.i.d. r.v.s with d.f. F n (x), X is a r.v. with F(x), if for all continuous points of F(x) we have It is said that {X n } convergence to X in distribution and denoted it by

2. Central Limit Theorems (CLT) Levy-Lindeberg’s CLT Suppose that {X n } are i.i.d. r.v.s with mean  <  and variance  2 <  ， k=1, 2, …, then {X n } follows the CLT, which also means that

Suppose that Z n (n=1, 2,...) follow binomial distribution with parameters n, p(0<p<1), then De Moivre-Laplace’s CLT Proof

Example 2 A life risk company 寿险公司 has received policies 保单, assume each policy with premium 保 险费 12 dollars and mortality rate 死亡率 0.6% ， the company has to paid 1000 dollars when a claim arrived, try to determine ： (1) the probability that the company could be deficit 亏 损？ (2)to make sure that the profit 利润 of the company is not less than dollars with probability 0.9, try to determine the most payment of each claim.

Let X denote the death of one year, then, X~B(n, p), where n= ， p=0.6% ， Let Y represent the profit of the company, then, Y=10000  X. By CLT, we have (1)P{Y<0}=P{10000  X<0}=1  P{X  120}  1   (7.75)=0. (2) Assume that the payment is a dollars, then P{Y>60000}=P{10000  12-X>60000}=P{X  60000/a}  0.9. By CLT, it is equal to

Abraham de Moivre Born: 26 May in Vitry (near Paris), France Died: 27 Nov in London, England

Pierre-Simon Laplace Born: 23 Mar in Beaumont-en-Auge, Normandy, France Died: 5 Mar in Paris, France