1. § 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. 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

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

4. 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 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.

5. 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 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