高等数学概率论与数理统计《初等概率论》PPT课件-第5讲.pdf

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1、5?V615CC?5l.CY.C!C1.1?m,uA,5(indicator function)IA?=IA()=1,?A,0,?/A.?x R,IA()v:|IA()x=,?x 0,Ac,?x 0,1),?x 1.ux?IA xdef=|IA()x.:?:k:5:?:C.15?V615CC?5l.CY.C!CduIA?1$C?5NBXK1.1y(A 4 B)4 C=A 4(B 4 C),A,B,C,4L?$.y.5?I(A4B)4C=|IA4B IC|=|IA IB|IC|,IA4(B4C)=|IA IB4C|=|IA|IB IC|.XJIB=0,KI(A4B)4C=|IA IC|=IA4(

2、B4C).XJIB=1,KI(A4B)4C=|1 IA IC|=IA4(B4C).25?V615CC?5l.CY.C!C1.23B?52?1?m?z:L1.XL?.XX=3L?3XX?m?=X:7 NKX=3def=|X()=3=3,3,3,3.w,x R,X x.XC.duX?X?1$.1.3 x=X xc F;2X x=n=1X x 1n F;3X x=X xc F.65?V615CC?5l.CY.C!C1.4352?1313Tk5rs?V.).XL/13rs?0KX=5%?P(X=5)=?135?398?/?5213?.?X()?m=,=d?5213?:|z?:gS?13.X()3?X(

3、)=“?rs”,.X=5=|k5rs.+X?XL5?KP(X=5)=?)vk?.d x F;2X x F;3X x F.?EC?1oK$aX+bY,XY,X/Y,a,b R;24supnXn;infnXn;limsupnXn;liminfnXn;limnXn(XJ3);3C:maxX,Y,minX,Y,sin(X),eX,X+,X,|X|p?;4E.85?V615CC?5l.CY.C!CRL?N,SLRmm4?fm?N=S=(a,b.-B=(S),B:BorelB?:Borel8.?(,F,P)Vme?nL?Borel8A,X AuOVP(X A).n1.1?X m(,F)?C K?Borel

4、8AkX Adef=|X()A F.y.A R,-X1(A)=|X()A,A=A|X1(A)F,A R.eyA-,B A.(a).X1(R)=|X()R=F,?R A;95?V615CC?5l.CY.C!C(b).?A A,kX1(A)F,uX1(Ac)=|X()Ac=|X()Ac=X1(A)c F.?Ac F.(c).Aj A,kX1(Aj)F,j=1,2,.,l?X1(i=1Ai)=|X()i=1Ai=i=1|X()Ai=i=1X1(Ai)F.?i=1Ai A.dA-.(a,b S,X1(a,b)=a XJX?01VP(X=1)=p=1 P(X=0),KXl:(Bernoulli)PX B

5、(1,p).?(!xG?!?!G?):?CX=1,?,0,?.?X01P1 pp.175?V615CC?5l.CY.Cn!l.C2?B(n,p)XJCX?VXeP(X=k)=?nk?pk(1 p)nk,k=0,1,.,n,KXl?p (0,1)PX B(n,p).?nk?pk(1 p)nk?m1=p+(1 p)n=nXk=0?nk?pk(1 p)nk?1k+1.BBinomial?i1.185?V615CC?5l.CY.Cn!l.C?S?Vp,?S-En,XL?gKX B(n,p).)AjL1jg?KA1,A2,.,AnpP(Aj)=p.lng?kg?k?nk?.1jA?j1,j2,.,jk,

6、Bj=Aj1 Aj2 Ajk Acjk+1 AcjnL1j1,j2,.,jkg?KBjpNX=k=(nk)j=1Bj,P(Bj)=pk(1 p)nk.|V?k5?P(X=k)=(nk)Xj=1P(Bj)=?nk?pk(1 p)nk.195?V615CC?5l.CY.Cn!l.CGGGGGGGGGGGGGGGGGGG0510150.00.10.20.30.4GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGn=3n=6n=9n=12n=15n=18:?p=0.5.205?V615CC?5l.CY.Cn!l.CGGGGGGGGGGGGGGGGGGG

7、GGGGGGGGGGGG0510152025300.00.10.20.30.40.50.6GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGn=3n=6n=9n=12n=15n=30:?p=0.2.215?V615CC?5l.CY.Cn!l.CGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG0510152025300.00.10.20.30.40.50.6GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGn=3n=6n=9n=12n=15n=30:?p=0.8.225?V615CC

8、?5l.CY.Cn!l.C-b(k,n,p)=P(X=k).Kb(k,n,p)b(k 1,n,p)=(n k+1)pk(1 p)=1+(n+1)p kk(1 p).?1?k b(k 1,n,p)d?ucb(k,n,p)k?O?O;2?k (n+1)pKb(k,n,p)b(k 1,n,p)d?ucb(k,n,p)k?O?;3?k=(n+1)pKb(k,n,p)=b(k1,n,p)db(n+1)p,n,p)b(n+1)p 1,n,p)?.235?V615CC?5l.CY.Cn!l.Cn3.1?Uk03=vb(k0,n,p)=max0knb(k,n,p)?k03k0=(n+1)p,(n+1)p 1,

9、XJ(n+1)p?,(n+1)p,XJ(n+1)p?,=(a)?(n+1)p?b(n+1)p,n,p)b(n+1)p 1,n,p)(b)?(n+1)p?b(n+1)p,n,p)?.b(k0,n,p)?%.245?V615CC?5l.CY.Cn!l.C3.198IlXJz?p?z?g?V0.3,k8I?V9U8I?.).?X8I?2)=9Xk=3?9k?(0.3)k(1 0.3)9k=1 2Xk=0?9k?(0.3)k(1 0.3)9k=1 0.4624=0.5376.q(n+1)p=10 0.3=3?U?8I?56265?V615CC?5l.CY.Cn!l.CGGGGGGGGGGGGGGGG0

10、510150.00.10.20.30.4GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG=1=2=3=4=5=6:Poisson?.275?V615CC?5l.CY.Cn!l.C285?V615CC?5l.CY.Cn!l.C3.21910cRutherfordGeiger*?5l(Polonium)?f?1?N=2608g*zg*7.5?*?10094 f.e*P.?CY P(3.87),3.87=10094/26087.5?f?.XL?5l37.5S?feL?LX=k3N=2608

11、g-E*u)?P(Y=k)?.295?V615CC?5l.CY.Cn!l.CLLL:?5l?f*?*?kfu)?P(Y=k)fk?gmkmk/N0570.0220.02112030.0780.08123830.1470.15635250.2010.20145320.2040.19554080.1560.15162730.1050.09771390.0530.0548450.0170.0269270.0100.01110+160.0060.007oO26080.9991.00305?V615CC?5l.CY.Cn!l.CGGGGGGGGGGG02468100.000.050.100.150.20

12、GGGGGGGGGGGmkNP(Y=k):f?mk/NVP(Y=k).315?V615CC?5l.CY.Cn!l.C eyX P().?t=7.5?nzn=t/n.?nb13nSkff?Vpn=n=t/n,p?2?mS?fp.3be?5?fX B(n,pn).uP(X=k)=limn?nk?pkn(1 pn)nk=(t)kk!n(n 1)(n k+1)nk?1 tn?nk=(t)kk!et.?=t,?X P().325?V615CC?5l.CY.Cn!l.C f?y?Poisson Cq?:XJnp?np?KP()Cq?B(n,p),AO?n(ATPoisson?.?A1,pcS?2,|U?g

13、3,U?4,U?oXJCX?VXeP(X=m)=?Mm?NMnm?Nn?,m=0,1,.,minn,M,KXlAPX H(n,M,N).?3N?oNM?N M?.?n?|.?|Tkm?V.A1u2-2?.3345?V615CC?5l.CY.Cn!l.C3.3N?TkMgl?nXLn?gKXlA.XJ1?k?vk?.?X B(n,pN)pN=M/Ng.S?limNM/N=pkP(X=m)=?Mm?NMnm?Nn?=M!m!(M m)!(N M)!(n m)!(N M n+m)!n!(N n)!N!=?nm?M(M 1)(M m+1)Nm(N M)(N M n+m+1)Nnm355?V615CC?

14、5l.CY.Cn!l.CNnN(N 1)(N n+1)?nm?pm(1 p)nm.=:A:?:C:q.3SKu?NO?Nn?,?Mm?,?NMnm?O?nm?.?n?NMCq?Mm?NMnm?Nn?nm?pmN(1 pN)nm B.365?V615CC?5l.CY.Cn!l.C5A G(p)XJCX?VXeP(X=k)=(1 p)k1p,k=1,2,.,KXlp?A.3.48I?.XLg8I?g.XJzg?V p 0KX G(p).y.AjL1jgv8IdAj?5?P(X=k)=P(A1 Ak1 Ack)=(1 p)k1p,k=1,2,.5?P(X k)=P(X=1).y.(=).dV?P(X

15、=k+1|X k)=P(X=k+1,X k)P(X k)=P(X=k+1)P?Sj=k+1X=j?=(1 p)kpPj=k+1(1 p)j1p=p=P(X=1).385?V615CC?5l.CY.Cn!l.C(=).?rk=P(X k),|X=k+1=X k X k+1,?p=P(X=k+1|X k)=P(X k)P(X k+1)P(X k)=1 rk+1rk.|r0=P(X 0)=1,?rk+1=(1 p)rk=.=(1 p)k+1.dP(X=k)=P(X k 1)P(X k)=rk1 rk=(1 p)k1p.395?V615CC?5l.CY.Cn!l.C6dk(Pascal distrib

16、ution)XJCX?VXeP(X=k)=?k 1r 1?(1 p)krpr,k=r,r+1,.,KXldk.?r=1dkA.3.58I?rg.XL?g.XJzg?V p (0,1)KXldk.405?V615CC?5l.CY.Cn!l.C7K?NB(r,p)=Polya XJCY?VXeP(Y=k)=?k+r 1r 1?(1 p)kpr,k=0,1,.,(1)KY NB(r,p).3f3.5,-Y=X rKY?”?g.3.6?u?,?p?k?Y?lT?eZu?XY?.,Ykg,r,/lT?u?uy1r.XP?u?.w,ep?KXu?KXu?.?Xup?8?.415?V615CC?5l.CY.

17、Cn!l.C 5?(a)./K?m0(1 x)r=Xk=0(r)(r 1)(r k+1)k!(x)k=Xk=0(r+k 1)(r+k 2)(r+1)rk!xk=Xk=0?k+r 1r 1?xk,|x|pr?1=pr1 (1 p)r=Xk=0?k+r 1r 1?(1 p)kpr.?y?(1)(?.425?V615CC?5l.CY.Cn!l.C(b).?.?/?10?eo?n?rX CK?Ker?ro?g?rC.435?V615CC?5l.CY.Co!Y.CY.C!V?CXXJ:3:K:f(x)?a b,P(a X b)=Zbaf(x)dx,KXY.C(continuous r.v.)f(x)X?

18、VV(density).?f(x)X?VK1Rf(x)dx=1.y.An=X (n,nNO|V?Y5?Zf(x)dx=limnZnnf(x)dx=limnP(An)=P?n=1An?=P(X (,)=1.445?V615CC?5l.CY.Co!Y.C2P(X=a)=0,uP(a X b)=P(a X b).y.|V?Y5?P(X=a)P(X (a ,a)=Zaaf(x)dx 0,as 0.3?Borel8A,kP(X A)=RAf(x)dx.,XY.C?XX=X()YX?mvk?(?!?Y5Y.C?g?Y.455?V615CC?5l.CY.Co!Y.C 3S7k.4.1f(x)=12x,x (

19、0,1.Kf(x)?f(x)3(0,1.limx0f(x)=+.465?V615CC?5l.CY.Co!Y.C?Y.C1!U(a,b)a bXJX?Vf(x)=1ba,x (a,b),0,x/(a,b),Xlm(a,b)?!PX U(a,b).UUniform?i1.w,m(a,b)?(a,b,a,b)a,b.f(x)?5?/f(x)=1b aI(a,b)(x)f(x)=1b aIx(a,b).475?V615CC?5l.CY.Co!Y.C?Borel8A,XJA?m(A)=RAdx ,aq/A?!XJX?f(x)=1m(A),x A,0,x Ac,XlBorel8A?!PX U(A).?Bo

20、rel8B,P(B)=m(A B)m(A).485?V615CC?5l.CY.Co!Y.C2E()?XJX?f(x)=ex,x 0,0,x 0,Xl?PX E().EExponential?i1.f(x)?5?/f(x)=exI0,)f(x)=ex,x 0.?CX?P(X s+t|X s)=P(X t).(2)5(2)?5P5.P5?A?.y.(=).?X E(),KP(X x)=Zxeydy=exV?P(X s+t|X s)=P(X s+t,X s)P(X s)=P(X s+t)P(X s)=e(s+t)es=et=P(X t).515?V615CC?5l.CY.Co!Y.C(=).?Xk5

21、(2),-G(x)=P(X x).dP(X s+t|X s)=P(X s+t)P(X s)=P(X t).?G(s+t)=G(s)G(t)=logG(s+t)=logG(s)+logG(t).duG(x)NO?.dp?logG(x)=logG(1)x,i.e.,G(x)=ex,=logP(X 1)0.?0 a b,P(a 0s.”?mS”?V.?Pz?/Pz0U?Cq.?3?Pzy?.3?(/?.X?35060cdu)nPz?k?g?e?0,m 0.535?V615CC?5l.CY.Co!Y.C3WeibullXJX?f(x;,)=x1exp(x),x 0,XlWeibull.(a)?=1We

22、ibull (b)?=2Weibull Rayleighf(x;)=x2exp?x222?,x 0,Weibull3:5:-/.545?V615CC?5l.CY.Co!Y.C4?N(,2)XJX?f(x)=12exp?(x )222?,x R,(3)Xl(,2)?PX N(,2).NNormal?i1.AO/?X N(0,1)XlIO?.IO?:(x)=12exp?x2/2?,x R,555?V615CC?5l.CY.Co!Y.C?dGauss3?qGauss.3Brown motion?.?3VnOkXA?/.L?NI)?N)nI?lCql?.pk?C?Cql?.?.q.ke51f(x)ux

23、=2f()=(2)13f(x)3x=?k$:.565?V615CC?5l.CY.Co!Y.C420240.00.20.40.60.8xf(x)=1=0.5=2:?/.575?V615CC?5l.CY.Co!Y.C(x)(.?12Zex2/2dx?12Zey2/2dy?=12ZZe(x2+y2)/2dxdy=12Z20dZ0rer2/2dr=12Z0er2/2dr2=1.(3)?f(x).Z12exp?(x )222?dx=12Zex2/2dx=1.585?V615CC?5l.CY.Co!Y.C?B(x)=Zx(t)dt.|(t)?5?(x)+(x)=1,(x)=1 (x),x R.X N(,2

24、),kP(X a)=12Zaexp?(x )222?dx=12Z(a)/ey2/2dy=?a?.595?V615CC?5l.CY.Co!Y.C?X N(,2)P(|X|)=(1)(1)=2(1)1=68.27%;P(|X|2)=(2)(2)=2(2)1=95.45%;P(|X|3)=(3)(3)=2(3)1=99.73%;P(|X|6)=2(6)1=99.999999802682%.3?+n?:36+:n:?:K?!?5U?5X?P(|X|3)P(|X|6).605?V615CC?5l.CY.Co!Y.C5Gamma(,)?,?XJX?f(x)=()x1ex,x 0,Xl(,)?GammaPX

25、 (,),()=Z0 x1exdx.(a)(n)=(n 1)!;(b)(1/2)=.?/G(shape parameter);?the rate parameter;?Erlang(3/0).615?V615CC?5l.CY.Co!Y.C=IOKarl Pearson3n!)9L?CuyY.C?.?C?A:?KuuaC?Wikipedia?/Pearson distribu-tion03Z/?c!G?Y?lmS?l.?/Pearson distribution0(Wikipedia).625?V615CC?5l.CY.Co!Y.C6BetaB(,)?,?XJX?f(x)=1B(,)x1(1 x

26、)1,0 x 0XJX?f(x)=12/2(/2)x21ex2,x 0,Xl?2PX 2.635?V615CC?5l.CY.Co!Y.C8Students t?0XJX?f(x)=?+12?2?1+x2?+12,x R,Xl?StudentPX t.9Cauchy XJX?f(x)=1(1+x2),x R,XlCauchyw,PX t1.645?V615CC?5l.CY.Co!Y.C:?KlC?:C?;,?Maria GaetanaAgnesi(1718/5/16-1799/1/9,|o)?.10F(m,n)=FisherSnedecor distribution?m,n?XJX?f(x)=?m+n2?mm/2nn/2xm21?m2?n2?(mx+n)m+n2,x 0,Xl(m,n)?FPX F(m,n).65