1、Y1?C1 YZ(Statistics,ECNU)1?C1/20?二?下YYI?I!I I Z(Statistics,ECNU)1?C2/20Y?X N(,2)p(x)=12exp(x )222),x 0,?.|-?5gCx=y+,y=2t,Zp(x)dx=Z12exp(x )222)dx=22Z0exp y22!dy=22Z0et22t1/2dt=1 12!=1.Z(Statistics,ECNU)1?C3/20Y?X N(,2)p(x)=12exp(x )222),x 0,?.|-?5gCx=y+,y=2t,Zp(x)dx=Z12exp(x )222)dx=22Z0exp y22!dy=2
2、2Z0et22t1/2dt=1 12!=1.Z(Statistics,ECNU)1?C3/20Y?V?5(1)p(x)ux=,3?;(2)eC,UC,Kp(x)”xm,?/G?C;(3)eC,UC,Kp(x)”C,?u)UC.Z(Statistics,ECNU)1?C4/20YIO?=0,=1?.X N(0,1).PIO?V(x),(x).(0)=12,(x)+(x)=1.Z(Statistics,ECNU)1?C5/20Y(x)?O(1)(0)=12;(2)x 0,?IO?L;(3)x 1.96)9P(|X|1.96)=1 P(X 1.96)=1 (1.96)=(1.96)=0.9750,P
3、(|X|1.96)9P(|X|1.96)=1 P(X 1.96)=1 (1.96)=(1.96)=0.9750,P(|X|1.96)=(1.96)(1.96)=2(1.96)1=0.95.Z(Statistics,ECNU)1?C7/20YExample 2?X N(0,1),P(X b)=0.9515,P(X a)=0.0505.a,b.(b)=0.9515,?IO?L,b=1.66.(a)=0.0505 1/2,ud(a)=1 (a)=1 0.0505=0.9495,?IO?L,?a=1.64,?a=1.64.Z(Statistics,ECNU)1?C8/20YExample 2?X N(
4、0,1),P(X b)=0.9515,P(X a)=0.0505.a,b.(b)=0.9515,?IO?L,b=1.66.(a)=0.0505 1/2,ud(a)=1 (a)=1 0.0505=0.9495,?IO?L,?a=1.64,?a=1.64.Z(Statistics,ECNU)1?C8/20Y?IOzTheorem 3?X N(,2),Y=X ,KY N(0,1).Z(Statistics,ECNU)1?C9/20Yyw,X?VpX(x)=12exp(x )222),x .?Y?FY(y),KFY(y)=P(Y y)=P?X y?=P(X +y)=Z+ypX(x)dx=ZypX(+t
5、)dt,?COx=+t.u,Y?VpY(y)=dFY(y)dy=ddy ZypX(+t)dt!=pX(+y)=12ey2/2.=pY(y)=(y)IO?N(0,1)?V,?Y N(0,1).Z(Statistics,ECNU)1?C10/20YCorollary 4?X N(,2),KX?F(x)=?x?.dK?n3,x N(0,1).?d?,X?F(x)=P(X x)=P?X x?=P?Y x?=?x?.Z(Statistics,ECNU)1?C11/20YCorollary 4?X N(,2),KX?F(x)=?x?.dK?n3,x N(0,1).?d?,X?F(x)=P(X x)=P?X
6、 x?=P?Y x?=?x?.Z(Statistics,ECNU)1?C11/20YExample 5?X N(10,4),VP(10 X 13)P(|X 10|2).d4,VP(10 X 13)=13 102!10 102!=(1.5)0.5=0.9332 0.5=0.4332,P(|X 10|2)=P(8 X 12)=12 102!8 102!=(1)(1)=2(1)1=2 0.8413 1=0.6826.Z(Statistics,ECNU)1?C12/20YExample 5?X N(10,4),VP(10 X 13)P(|X 10|2).d4,VP(10 X 13)=13 102!10
7、 102!=(1.5)0.5=0.9332 0.5=0.4332,P(|X 10|2)=P(8 X 12)=12 102!8 102!=(1)(1)=2(1)1=2 0.8413 1=0.6826.Z(Statistics,ECNU)1?C12/20YExample 6?X N(,2),P(X 5)=0.063 P(X 3)=0.6179.,9P(|X|).dK4,5 !=F(5)=0.063,3 !=F(3)=0.6179.5?(x)+(x)=1,?L?,5+=1.53,3 =0.3.)?=1.69,=4.37.dn3,P(|X|)=P?X?1?=2(1)1=0.6826.Z(Statist
8、ics,ECNU)1?C13/20YExample 6?X N(,2),P(X 5)=0.063 P(X 3)=0.6179.,9P(|X|).dK4,5 !=F(5)=0.063,3 !=F(3)=0.6179.5?(x)+(x)=1,?L?,5+=1.53,3 =0.3.)?=1.69,=4.37.dn3,P(|X|)=P?X?1?=2(1)1=0.6826.Z(Statistics,ECNU)1?C13/20Y?3?K?X N(,2),KI P(|X|)=0.6828;I P(|X|2)=0.9545;I P(|X|3)=0.9973.L,XA?o3(3,+3)S?,?3OOOKKK.O
9、K?2/A?+n,pS.u6+nOK.Z(Statistics,ECNU)1?C14/20Y?3?K?X N(,2),KI P(|X|)=0.6828;I P(|X|2)=0.9545;I P(|X|3)=0.9973.L,XA?o3(3,+3)S?,?3OOOKKK.OK?2/A?+n,pS.u6+nOK.Z(Statistics,ECNU)1?C14/20Y!X U(a,b)p(x)=1b a,a x b;0,K.F(x)=0,x a;x ab a,a x b;1,b x.Z(Statistics,ECNU)1?C15/20YExample 7?X U(2,5).y3X?1ng*?kg*u
10、3?V.X?Vp(x)=13,2 x 3)=R3p(x)dx=R5313dx=23.?YLX?1ng*,*u3?g,KY b 3,23!.uP(Y 2)=1 P(Y=0)P(Y=1)=1 1 23!3 3 23!1 1 23!2=2027=V.Z(Statistics,ECNU)1?C16/20YExample 7?X U(2,5).y3X?1ng*?kg*u3?V.X?Vp(x)=13,2 x 3)=R3p(x)dx=R5313dx=23.?YLX?1ng*,*u3?g,KY b 3,23!.uP(Y 2)=1 P(Y=0)P(Y=1)=1 1 23!3 3 23!1 1 23!2=2027
11、=V.Z(Statistics,ECNU)1?C16/20YX Ga(,),p 0,0.p(x)=()x1ex,x 0;0,x 0.()=Z0 x1exdxGamma,(1)=1,(1/2)=,(s+1)=s(s).Z(Statistics,ECNU)1?C17/20Y5PI Ga(1,)=Exp();I Ga n2,12!=2(n).Z(Statistics,ECNU)1?C18/20Y5PI Ga(1,)=Exp();I Ga n2,12!=2(n).Z(Statistics,ECNU)1?C18/20YX Exp(),p 0.p(x)=ex,x 0;0,x 0.F(x)=1 ex,x 0
12、;0,x 0.A2,F)?F?,?(X?!?nA?)3?ml.Z(Statistics,ECNU)1?C19/20YX Exp(),p 0.p(x)=ex,x 0;0,x 0.F(x)=1 ex,x 0;0,x 0.A2,F)?F?,?(X?!?nA?)3?ml.Z(Statistics,ECNU)1?C19/20YkP5,=P(X s+t|X s)=P(X t).X?F(x)=Zxp(t)dt=Zx0etdt,x 0,0,x 0.=1 ex,x 0,0,x 0.u,?s,t 0,P(X s+t|X s)=P(X s+t,X s)P(X s)=1 F(s+t)1 F(s)=1 (1 e(s+t
13、)1 (1 es)=et=1 F(t)=P(X t).Z(Statistics,ECNU)1?C20/20YkP5,=P(X s+t|X s)=P(X t).X?F(x)=Zxp(t)dt=Zx0etdt,x 0,0,x 0.=1 ex,x 0,0,x 0.u,?s,t 0,P(X s+t|X s)=P(X s+t,X s)P(X s)=1 F(s+t)1 F(s)=1 (1 e(s+t)1 (1 es)=et=1 F(t)=P(X t).Z(Statistics,ECNU)1?C20/20iA?1?C18 iA?Z?Z(Statistics,ECNU)1?C1/11iA?Definition
14、 1?lCXk?P(X=xk)=pk,k=1,2,.e?Xk=1xk pk,KXk=1xk pkX?,PEX.=EX=Xk=1xk pk.Z(Statistics,ECNU)1?C2/11iA?Definition 2?YCXkp(x).eZx p(x)dx,KZx p(x)dx X?,PEX.=EX=Zx p(x)dx.5P:q,k,?.Z(Statistics,ECNU)1?C3/11iA?Definition 2?YCXkp(x).eZx p(x)dx,KZx p(x)dx X?,PEX.=EX=Zx p(x)dx.5P:q,k,?.Z(Statistics,ECNU)1?C3/11iA
15、?Example 3Xk?X-1012P0.20.10.40.3OX?.EX=1 0.2+0 0.1+1 0.4+2 0.3=0.8.Z(Statistics,ECNU)1?C4/11iA?Example 3Xk?X-1012P0.20.10.40.3OX?.EX=1 0.2+0 0.1+1 0.4+2 0.3=0.8.Z(Statistics,ECNU)1?C4/11iA?Example 4O?b(n,p)?.CXl?b(n,p),K?P(X=k)=Cknpk(1 p)nk,k=0,1,.,n.d?,EX=XkkP(X=k)=nXk=0kCknpk(1 p)nk=nXk=1k n!k!(n
16、k)!pk(1 p)nk=npnXk=1(n 1)!(k 1)!(n 1 (k 1)!pk1(1 p)(n1(k1)=npn1Xj=0(n 1)!j!(n 1 j)!pj(1 p)n1j=np,?A?b(n 1,p)?K5.Z(Statistics,ECNU)1?C5/11iA?Example 4O?b(n,p)?.CXl?b(n,p),K?P(X=k)=Cknpk(1 p)nk,k=0,1,.,n.d?,EX=XkkP(X=k)=nXk=0kCknpk(1 p)nk=nXk=1k n!k!(n k)!pk(1 p)nk=npnXk=1(n 1)!(k 1)!(n 1 (k 1)!pk1(1 p)(n1(k1)=npn1Xj=0(n 1)!j!(n 1 j)!pj(1 p)n1j=np,?A?b(n 1,p)?K5.Z(Statistics,ECNU)1?C5/11iA?Example 5OExp()IO?N(0,1)?.?CXlExp(),KVp(x)=ex,x 0,0,x 0.d?,EX=Zxp(x)dx=Z0 x exdx=1.?YlIO?N(0,1),KY?V(y)=12ey2
