S 5.3. Central Limit Theorems1. Convergence in distributionSuppose that Xn) are i.i.d. r.v.s with d.f. Fn(x), X is ar.v. with F(x), if for all continuous points of F(x) we havelim Fn(x) = F(x),n-oIt is said that (X,) convergence to X in distributionand denoted it byXnwx.1Set Y,- Xk, denoted the standardized Y, byk=1then Y*w→~ N(O, 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 lim 𝑛→∞ 𝐹𝑛(𝑥) = 𝐹(𝑥), It is said that {Xn } convergence to X in distribution and denoted it by 𝑋𝑛𝑤𝑋. * 1 * Set , denoted the standardized by , then ~ (0,1), n n k n n k w n Y X Y Y Y N = = ⎯⎯→
2. Central Limit Theorems (CLT)Levy-Lindeberg' s CLTSuppose that [X,i are i.i.d. r.v.s with mean μ<ooand variance 2<oo , k-1, 2, ..., then {X,j followsthe CLT. which also means thatipXi ≤xDInd(2r,X, - nμlim Fn(x) = limgynn-8n-212元=Φ(x)
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 𝑝 𝑖=1 𝑛 𝑋𝑖 ≤ 𝑥 ≈ 𝛷( 𝑥 − 𝑛𝜇 𝑛𝜎 ) lim 𝑛→∞ 𝐹𝑛(𝑥) = lim 𝑛→∞ 𝑃 σ𝑖=1 𝑛 𝑋𝑖 − 𝑛𝜇 𝜎 𝑛 ≤ 𝑥 = න 1 2𝜋 𝑒 −𝑡 2/2dt = 𝛷(𝑥)
De Moivre-Laplace's CLTSuppose that Z, (n-1, 2, ...) follow binomial distribution withparameters n, p(0<p<1), thenZn-nplim P!e 2 dt =Φ(x)≤x/=12元n-0/np(1- p)ProofZ, = ZXk, Xk : B(1, p)k=1E(X)= p, D(X)= p(1- p)ZX,-npZ,-np≤ x} = lim P) k=llim P≤x) =Φ(x)n00n-00/np(1- p)/np(1- p)
Suppose that Zn (n=1, 2, .) follow binomial distribution with parameters n, p(0<p<1), then De Moivre-Laplace’s CLT 2 2 1 lim { } ( ) (1 ) 2 t x n n Z np P x e dt x np p − → − − = = − Proof 1 , (1, ) n n k k k Z X X B p = = : ( ) , ( ) (1 ) E X p D X p p k k = = − 1 lim { } lim { } ( ). (1 ) (1 ) n k n k n n X np Z np P x P x x np p np p = → → − − = = − −
Example2Aliferiskcompany寿险公司hasreceived10000 policies保单,assume each policy with premium保险费 12 dollars and mortality rate死亡率 0.6%，thecompany has to paid 1000 dollars when a claim arrived,try to determine:(1) the probability that the company could be deficit 号损？(2)to make sure that the profit利 润 of the company isnot less than 60000 dollars with probability 0.9, try todetermine the most payment of each claim
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)wheren=10000,p=0.6%,LetYrepresenttheprofitofthecompany, then, Y=10000x12-1000X. By CLT, we have(1)P[Y<0)=P[10000×12-1000X<0)=1-P[X<≤120]~1 - Φ(7.75)=0.(2) Assume that the payment is a dollars, thenP^Y>60000)=P10000x12-X>60000)=PX≤60000/a)≥0.9ByCLT, itis equal to6000010000 × 0.006ad) ≥ 0.9/10000×0.006×0.994=a≤3017
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=1000012-1000X. By CLT, we have (1)P{Y<0}=P{1000012-1000X<0}=1−P{X120} 1 − (7.75)=0. (2) Assume that the payment is a dollars, then P{Y>60000}=P{1000012-X>60000}=P{X60000/a}0.9. By CLT, it is equal to 𝛷( 60000 𝑎 − 10000 × 0.006 10000 × 0.006 × 0.994 ) ≥ 0.9 ⇒ 𝑎 ≤ 3017