1、1n V1?C1n VZ(Statistics,ECNU)1?C1/30?二?1n Vl.eCX?U?x1,x2,xn,pk=P(X=xk),k=1,2,X?,Xl.Ckl.?L/L.Xx1x2xnPp1p2pnZ(Statistics,ECNU)1?C2/301n Vl.eCX?U?x1,x2,xn,pk=P(X=xk),k=1,2,X?,Xl.Ckl.?L/L.Xx1x2xnPp1p2pnZ(Statistics,ECNU)1?C2/301n VExample 1:eCXvP(X=c)=1,=X?Fz,KXl:,PX c,?XcP1.Z(Statistics,ECNU)1?C3/301n
2、VExample 2?qM1,=H,T,F=,H,T,VPvP(H)=1/3.uCX()=1,e=H;0,e=T.?X01P2/31/3.Z(Statistics,ECNU)1?C4/301n V5PlC?A5:I(C?kU?;I Oz?:?V.Z(Statistics,ECNU)1?C5/301n V?5:(1)K5:pk 0,k=1,2,(2)?K5:Xk=1pk=1.5ek?vK5?K5,K7,l.C?.Z(Statistics,ECNU)1?C6/301n V?5:(1)K5:pk 0,k=1,2,(2)?K5:Xk=1pk=1.5ek?vK5?K5,K7,l.C?.Z(Statist
3、ics,ECNU)1?C6/301n VTheorem 3?l.CXk?pk=P(X=xk),k=1,2KX?F(x)=Xk:xkxpk,p?Pkpk=0.I5?=PkX=xk,?x,X x=X x =Xk:xkxX=xk,d?V?5=?.Z(Statistics,ECNU)1?C7/301n VTheorem 3?l.CXk?pk=P(X=xk),k=1,2KX?F(x)=Xk:xkxpk,p?Pkpk=0.I5?=PkX=xk,?x,X x=X x =Xk:xkxX=xk,d?V?5=?.Z(Statistics,ECNU)1?C7/301n V/,?D R,CXk?pk:k 1,KP(X
4、 D)=Xk:xkDpk.Z(Statistics,ECNU)1?C8/301n VExample 4CX?X012P1/21/31/6,X?VP(0 X 5/2).Z(Statistics,ECNU)1?C9/301n Vdn3,X?F(x)=Xk:xkxpk=0,x 0;12,0 x 1;56,1 x 2;1,2 x.VP 0 X 52!=Xk:xk(0,5/2)pk=P(X=1)+P(X=2)=13+16=12.Z(Statistics,ECNU)1?C10/301n VExample 5CX?P(X=k)=23 13!k1,k=1,2,VP(X 2).P(X 2)=1 P(X 2)=1
5、 P(X=1)=1 23=13.Z(Statistics,ECNU)1?C11/301n VExample 5CX?P(X=k)=23 13!k1,k=1,2,VP(X 2).P(X 2)=1 P(X 2)=1 P(X=1)=1 23=13.Z(Statistics,ECNU)1?C11/301n Vl.C?A?F(x)lCX?,KF(x)(1)N?F;(2)3m:?mY?;(3)m:=X?U?:;(4)3m:?a?pA?V.Z(Statistics,ECNU)1?C12/301n V,dlC?A?,?:Theorem 6?F(x)l.CX?,KX?U?:F?km:x1,x2,?P(X=xk)
6、=F(xk)F(xk 0),k=1,2,Z(Statistics,ECNU)1?C13/301n VExample 7X?F(x)=0,x 0;0.4,0 x 10.8,1 x 21,x 2X?.Z(Statistics,ECNU)1?C14/301n VX?U?:F?m:,=0,1,2.P(X=0)=F(0)F(0 0)=0.4 0=0.4;P(X=1)=F(1)F(1 0)=0.8 0.4=0.4;P(X=2)=F(2)F(2 0)=1 0.8=0.2P(X=2)=1 P(X=0)P(X=1)=1 0.4 0.4=0.2.?X?X012P0.40.40.2.Z(Statistics,ECN
7、U)1?C15/301n VY.Definition 8?CX?F(x).e3p(x)?x,F(x)=Zxp(t)dt,KXYC.p(x)V.Z(Statistics,ECNU)1?C16/301n VExample 9?CXl,=F(x)=1?arctanx+2?,x .XY.C,Vp(x)=1(1+x2),x 0;0,x 0.(1)k;(2)F(x).(1)dV?K5,1=Zp(x)dx=Z0ke3xdx=k3,)?k=3.Z(Statistics,ECNU)1?C20/301n VExample 10?CX p(x)=ke3x,x 0;0,x 0.(1)k;(2)F(x).(1)dV?K
8、5,1=Zp(x)dx=Z0ke3xdx=k3,)?k=3.Z(Statistics,ECNU)1?C20/301n V(2)PD=(0,),d(1),X?Vp(x)=3e3x 1D(x)=3e3x,x 0,0,x 0.?x R,PDx=(,x,KDxD=1(,0(x)+(0,x1(0,)(x).u,dY.C?,F(x)=Zxp(t)dt=ZDxp(t)dt=ZDxD3e3tdt=Zx03e3tdt,x 0,0,x 0=1 e3x,x 0,0,x 0.Z(Statistics,ECNU)1?C21/301n VExample 11?CX p(x)=1+x,1 x 0;1 x,0 x 1;0,K
9、.F(x).Z(Statistics,ECNU)1?C22/301n VKey:F(x)=0,x 1;x22+x+12,1 x 0;x22+x+12,0 x 0;0,x 0.VP(|X|1).PD=(1,1),VP(|X|0;0,x 0.VP(|X|1).PD=(1,1),VP(|X|1)=ZDp(x)dx=ZD(0,)3e3xdx=Z103e3xdx=1 e3.Z(Statistics,ECNU)1?C25/301n VTheorem 14?CX?Vp(x),=?x,p(x)=p(x).u,?a,FvF(a)=12Za0p(x)dx,F(a)+F(a)=1.AO/,F(0)=1/2,P(|X
10、|a)=2F(a)1,P(|X|a)=2(1 F(a).Z(Statistics,ECNU)1?C26/301n Vn14?yp(x),?dV?K5,1=Zp(y)dy=2Z0p(y)dy=2Z0p(y)dy=2F(0).?F(0)=1/2,Z0p(y)dy=1/2.dY.C?,?a,F(a)=Zap(x)dx=Zap(x)dx=Zap(y)d(y)=Zap(y)dy=Z0p(y)dy Za0p(y)dy=12Za0p(x)dx.Z(Statistics,ECNU)1?C27/301n Vn14?yYqdZap(y)dy=Zp(y)dy Zap(y)dy=1 F(a),?F(a)+F(a)=1
11、.uP(|X|a)=P(X a)P(X a)=F(a)F(a)=2F(a)1,P(|X|a)=1 P(|X|a)=1 (2F(a)1)=2(1 F(a).Z(Statistics,ECNU)1?C28/301n V.Definition 15?F1(x),l.C?,F2(x),Y5C?,Ny?:0 1,F(x)=F1(x)+(1 )F2(x),A?.Z(Statistics,ECNU)1?C29/301n VExample 16?,OY?oYX,FX(x)=0,x 0,x3,0 x m+n|X m)=P(X n)?K?m,n.dVN?yP(X m+n|X m)=P(X n)=(1 p)n.P5
12、L:3?|?,3cmg?e,?e?ng?E?V”?gm.d?dp?,kP5?l7AZ(Statistics,ECNU)1?C8/18lP5Theorem 4?X Ge(p),KP(X m+n|X m)=P(X n)?K?m,n.dVN?yP(X m+n|X m)=P(X n)=(1 p)n.P5L:3?|?,3cmg?e,?e?ng?E?V”?gm.d?dp?,kP5?l7AZ(Statistics,ECNU)1?C8/18lP5Theorem 4?X Ge(p),KP(X m+n|X m)=P(X n)?K?m,n.dVN?yP(X m+n|X m)=P(X n)=(1 p)n.P5L:3?
13、|?,3cmg?e,?e?ng?E?V”?gm.d?dp?,kP5?l7AZ(Statistics,ECNU)1?C8/18lP5Theorem 4?X Ge(p),KP(X m+n|X m)=P(X n)?K?m,n.dVN?yP(X m+n|X m)=P(X n)=(1 p)n.P5L:3?|?,3cmg?e,?e?ng?E?V”?gm.d?dp?,kP5?l7AZ(Statistics,ECNU)1?C8/18lK?XL?|?1rg?o?g,KXk?P(X=k)=Cr1k1pr(1 p)kr,k=r,r+1,Xl(r,p)?KKK?dddkkk(Pascal),PX Nb(r,p).w,
14、Nb(1,p)=Ge(p).Z(Statistics,ECNU)1?C9/18l?K5?y:d?m,(1 x)r=Xj=0(r)(r 1)(r j+1)j!(x)j=Xj=0Cjj+r1xj,-x=1 p?,Xk=rpk=prXj=0Cr1j+r1(1 p)j=prXj=0Cjj+r1(1 p)j=pr(1 (1 p)r=1.Z(Statistics,ECNU)1?C10/18lExample 5(Banachs?K)=BanachUzU?z?kn?O3mpzg/l?guy?,ek(0 k n)?VZ(Statistics,ECNU)1?C11/18lI?g?wg?,?AL/?0,KP(A)=
15、1/2.I PXLA u)n+1g?g,KX Nb(n+1,1/2).I PBL/guy?,m?k0,KB=X=n+1+n k=2n k+1.Z(Statistics,ECNU)1?C12/18l?P(B)=P(X=2n k+1)=Cn+112nk+11 12!n+1 1 12!2nk+1(n+1)=Cn2nk 12!2nk+1.d5,/guym?,?k0?VCn2nk 12!2nk+1.?guy?,?,ek?V2 Cn2nk 12!2nk+1=Cn2nk 12!2nk,k=0,1.,n.Z(Statistics,ECNU)1?C13/18lteCXk?P(X=k)=kk!e,k=0,1,2,
16、KXl?t,PX P().?K5?y:3f(x)=ex?mex=Xk=0 xkk!-x=,?Xk=0pk=eXk=0kk!=e e=1.Z(Statistics,ECNU)1?C14/18lteCXk?P(X=k)=kk!e,k=0,1,2,KXl?t,PX P().?K5?y:3f(x)=ex?mex=Xk=0 xkk!-x=,?Xk=0pk=eXk=0kk!=e e=1.Z(Statistics,ECNU)1?C14/18lExample 6,zk:3(bt?:l3t?t).3:?V?.?XL3:,KX P(9),uX?P(X=k)=9kk!e9,k=0,1,2,.Z(Statistics,ECNU)1?C15/18lExample 6,zk:3(bt?:l3t?t).3:?V?.?XL3:,KX P(9),uX?P(X=k)=9kk!e9,k=0,1,2,.Z(Statistics,ECNU)1?C15/18lTheorem 7tn 3n-?p?Ppng?V.enpn,KCknpkn(1 pn)nkkk!e.?k,5?Cknpkn(1 pn)nk=(npn)kk!nnn 1nn
