1 / 5

§ 5.3 . Central Limit Theorems 1. Convergence in distribution

§ 5.3 . Central Limit Theorems 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).

keren
Télécharger la présentation

§ 5.3 . Central Limit Theorems 1. Convergence in distribution

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. § 5.3. Central Limit Theorems1. 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 10000 policies保单, assume each policy with premium保险费 12 dollarsand 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.

  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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

More Related