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

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

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

Example 2 A life risk company寿险公司 has received 10000 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 60000 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= 10000，p=0.6%，Let Y represent the profit of the company, then, Y=1000012-1000X. By CLT, we have (1)P{Y<0}=P{1000012-1000X<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