1、cI|:1009-4873(2007)04-0004-05几个著名大数定律的证明及应用 路庆华(石家庄信息工程职业学院基础部,河北石家庄 050035)K 1:大数定律以严格的数学形式表达了随机现象最根本的性质平均结果的稳定性,它是随机现象统计规律性的具体表现,介绍了几种常用大数定律及其证明方法,并分析了它们在理论和实际中的应用.1oM:大数定律;随机变量;数学期望;概率ms |:O212.2 DS M :A1 q d9 Cd9? p S,7 Cd9? pMHq/v k4C .V qd9l V A:B Yq? 3 q , “ kQ 9, Yq qv . Ll4 Bt C H,9?Cv 8 (r
2、T. , Y 8# kVY+ ,v 8 (rT B8+1, O . I n, 4“5: Mcl I ? I Hq/ ? v p15.2 +v p# v p -,5 +M1l:l1 !n(n =1,2,) q bW(,F ,P) l M (e ), i M , P i0,:limn p|n -|=0limn p|n -|=1,5 nG q l M (9 V B ),i/ |V U:limnn =(p)n p .l2 !nB , E(n)i, 7n =1n ni=1i , limn n -E(n) =0(P),5 nVv p, vE5 .l3 !Fn(x) sf , iBdf F(x), B x
3、,limnFn(x)=F(x),Fn(x) w F(x),5sf Fn(x) l F(x).l4 !Fn(x)(n =1,2,3,),F(x)sY M n(n =1,2,3,)#sf , Fn(x) w F(x),5nGs l ,g:n L , O:(1) n ,5n L ;(2)!c ,5n P c 1Hq n L c.IK : !+f fn(t) l Bf f(t), Of(t)t =0 H ,5Msf Fn(x) l Bsf F(x),7 Of(t) F(x)+f .1 T: ! B M , a,Z2,5 i :P|-a|22 , (1)P|-a|1 -nk/n22 =1 -k/n2,1
4、 P 1n ni=1i -1n ni=1ai 1 - kn2 . T |K,5limn P 1n ni=1i -1n ni =1ai a0 x =a,yN, 1n nk=1k L a,(2) T:1n nk=1kP a. 3 ( m v p)!n nQ k YqA? 3Q ,p YqA Q k? 3 q,5 i ,limn p nn -p 0,0 P| n - Pn |D( n)2 14n2.H |K,limn P| n - Pn|=0. v p m v pw 0,limn p 1n nk =1k - uW a,b , * ,iB TB1(x),B2(x), ,B l ff(x), x a,b
5、 . !a =0,b =1. V M u:x =(b -a)u +a, Pu 0,1“,L !f(x),x 0,1 f , * f(x)0,1 B i O. i0,0 x1, x2 0,1i0, P|f(x1)-f(x2)|0, P0 f(x)f(x)/c 0. Vnz 0 =Eg()0,#h(y ,z)(y0,z 0) : i0,i0,|y -y0 | |z - z0 | H, |h(y ,z)-h(y0,z 0)|.yN,P nn-Ef(1)Eg(1)=P|h(n ,n)-h(y0,z 0)|P|n -Ef(1)|,|n -Eg()|1-P|n -Ef(1)|-P|n -Eg(1)|.N
6、Vn:limnP nn -Ef(1)Eg(1) =1.limn1n ni=1f(i)1n ni=1g(i)=baf(x)dxbag(x)dx (P). ID:1 b C, . q d9 M .:v,2005. 2 d. q d9$ M .:,2003.3 I : On the evidence and application of a few well-famed laws of large numbersLU Qing-hua(Comprehensive Teaching Department,Shijiazhuang Information Technology Institute,Shijiazhuang,Hebei 050035,China)Abstract:This paper introduces the evidences of a few famed laws of large numbers,and their applicationsinpractice.Key words:laws of large numbers;random variable;mathematical expectation8 FE / 19
