$ 5.2 Law of large numbers1. Convergence in probabilitySuppose that (Xnj is a sequence of r.v.s, if for anyε>0, we havelimP(Xn -α<) =11-it is said that (X,j convergence to a in probabilityand denoted it byPX.-→a
§ 5.2 Law of large numbers 1. 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 a in probability and denoted it by lim { } 1 n n P X a → − = p X a n ⎯⎯→
PRemark X, →a Means whenn→8the probability that the value of X, fallin interval(a-,a +) is increased to 1.X.aa-εa+sX, →a meansV> 0,En.whenn>noIX,-ak?
X a P Remark n → Means when a− a a + Xn Xn → a means 0 0,n | X − a | n n → the probability that the value of Xn fall in interval (a −,a + ) is increased to 1. when n n0
2. Law of Large Numbers (LLN)1.Chebyshev's LLNSuppose that (Xk, k-1,2,..J are independent r.v.swith mean μ=E(Xk) and variance c2=D(Xk)>0 , thenZxμY. =二nk=li.e. for any give >, we havelim P([Y, - μ < } = 1n→0
2. Law of Large Numbers (LLN) 1. Chebyshev’s LLN Suppose that {Xk , k=1,2,.} are independent r.v.s with mean μ=E(Xk ) and variance σ 2=D(Xk)>0，then = ⎯→ = P n k n Xk n Y 1 1 i.e. for any give >0, we have lim {| − | } = 1 → n n P Y
Proof Chebyshev's inequality, we haveD(Y,)11P(/ Y, -E(Y,)K ε} ≥1-whereZE(Y.E(X,)= μnk=1naZD(Y,)D(X) :-二nnk=1thus92P(l Yn -μ<8}≥1-ne?lim P(l Yn -μ < } =1n→0
Proof Chebyshev’s inequality, we have . ( ) {| ( )| } 1 2 n n n D Y P Y − E Y − where = = = n k n E Xk n E Y 1 ( ) 1 ( ) n D X n D Y n k n k 2 1 2 ( ) 1 ( ) = = = {| | } 1 . 2 2 n P Yn − − thus lim {| − | } = 1 → n n P Y
2.Bernoulli's LLN-P95Set n(A) records the numbers of outcomesof A in n Bernoulli experiment, P(A)= p > O,then for any >O , we haven(A)lim P1<8n>00nProof X, : B(1, p),E(X) =p,D(X) = pq:n(A)=X, +X2 +L +Xn 2pqE(n(A)=- E(X,)= p, D( D(X)nnnnn i=li=1From Chebyshev's LLNn(A)lim P(Ipk=)=1.n→on
2.Bernoulli’s LLN-P95 Set n(A) records the numbers of outcomes of A in Bernoulli experiment, , then for any , we have n P A p ( ) 0 = 0 ( ) lim 1 n n A P p n → − = 1 2 2 1 1 Proof (1, ), ( ) , ( ) , ( ) , ( ) 1 ( ) 1 ( ) ( ) , ( ) ( ) , ' , ( ) lim (| | ) 1. i i i n n n i i i i n X B p E X p D X pq n A X X X n A n A pq E E X p D D X n n n n n From Chebyshev s LLN n A P p n = = → = = = + + + = = = = − = : L