概率与统计第十三讲随机变量的方差与协方差.ppt

Chapter 4 Limit Theorem * * * * § 4.1 Law of large number1. Convergence in probability Suppose that {Xn} is a sequence of r.v.s, if for any ?0, we have it is said that {Xn} convergence to X in probability and denoted it by Remark Means when a means the probability that the value of Xn is increased to 1 when fall in interval 2. Law of Large Numbers (LLN) 1. Chebyshev’s LLN Suppose that {Xk,k=1,2,...} are i.i.d r.v.s with mean ? and variance ?20，then i.e. for any give ?0, we have Proof Chebyshev’s inequality, we have where thus 2.Bernoulli’s LLN Set records the numbers of outcomes of A in Bernoulli experiment, , then for any ， we have 3. Khinchine’s LLN Suppose that {Xk,k=1.2,...} are i.i.d sequence with EXk=? ?, k=1, 2, … then Remark Suppose that {Xi,i=1.2,...} are i.i.d. r.v.s with E(X1k) =?, then This remark is very important for moment estimation for parameters to be discussed in Chapter 6. § 4.2. 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. 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 Suppose that ?n(n=1, 2, ...) follow binomial distribution with parameters n, p(0p1), then De Moivre-Laplace’s CLT Example 2 A life risk company has received 10000 policies, assume each policy with premium 12 dollars and mortality rate 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？ (2)to make sure that the profit of the company is not less than 60000 dollars with probability 0.9, try to determine the most payment of each claim. * *