概率论与数理统计小结

1、概率论与数理统计主要内容小结概率部分P() P(A| B )P(B ) P(A| B )P(B ) P(A| B )P(B )1122nnB ,B ,B S12nP(B )P(A| B )iP(B | )iinP(B )P(A| B )jjj1B ,B , ,BS12n A B ,P(A B) 0,B ,B P(AB) P()(B); x k p p) ,k X b pkkX b(n, p),x k C p p) ,k . kknkn e k X ( ),P x k ,kk!1,x(a,b) f(x) b aX U(a,b.其它1xe ,x 0X E ),f(x).其它x2e ,1 X N(

2、, ), .f(x)x22X ）F() 1，F() 0,X f(x)F(x) xft)；F(x) f(x) F(x).X f(x), x f(x)dxLX f (x),Y g(X)YX g(x)g (x) 0g (x) 0则 Yf h(y)|h (y)|,X yf (y) 其他Yh(y)是g(x) ming(g( maxg(g( y g(x)YF(y) Y g(X) X f (x)dxXB f (y) F(y).Y f (y) F (y) f(x)dx f( (y) (y) f( (y) (y)(y)Y(y)对于二维连续性随机变量(X,Y) f(x,yF(x,y fu,v)dvdu,则F(x,

3、y) xy ） f(x,y) fu,v) 1 f(x,y(x,y) f(x,y)dydx,D fu,v)dudv,F (x) x fu,v)dvdu, F (y) yXX f (x) f(x,y)dy, f (y) f(x,y).XY fu,y) f(x,v)F (x| y) XYdu, F (y| x) Y|Xdv,xyf (y)Yf (x)Xf(x,y)f(x,y) f (x| y) XY, f (y| x) Y|X.f (y)Yf (x)XX x ,Y y pijX x p p ，Y y p pii.j.jj1i1pppY y | X x ，P X x Y y|ijijpjiiji.j(

4、X,Y) f(x,y)F(x,y) 则Zf (z) Zf(z y,y)f(x,z x)当X,Y f (z) f (z y)f (y) f (x)f (zx)ZXYXY与当X,Y F (z) F (z)F (z) F (z) 1 F (z F (z)，MXYNXY f(x) E(X) (x)Y g(X), 则XEY) g(x)f(x).离散随机变量分布律为 x x pE(X) x pY g(X)， 则; 若, 则kkkkk1E(X) g(x )p .kkk1若 有二 维 的随 机 变 量 (X,Y) , 其 联合 概 率 密 度 为 f(x,y) ， 若 Y g(X,Y), 则 EY) g(x,

5、y)f(x,y). EC) C,E) (XE(X Y) E(X)EY)E(k X k X k X ) k E(X )k E(X )k E(X )1122nn1122nnE() E(X)EYX,YD(X) EX E(X) D(X) .2D(X) E(X ) E(X)22D C D X C( ) 0, ()( ), ( )D X D CX( ).C D X2D(X Y) D(X)DY)2E(X Y E(X) p,D(X) p pX b(n, p),E(X) np,D(X) np p X ( .E(X),D(X)abba)2,D(X) X U(a,bE(X).2X E( .2E(X),D(X) X

6、N( , ),E(X),D(X).22: Cov(X,Y) X E(XY EY E()E(X)EYCov(X,Y).(X,Y) D(X) DY) .XYXYD(X) DY)Cov(X,Y) CovY,X),Cov(X,X) D(X),Cov(X,a) 0Cov(,) abCov(X,Y),Cov(X X ,Y) Cov(X ,Y)Cov(X ,Y)1212D(X Y) D(X)DY)Cov(X,Y)X,Y D(X Y) D(X)DY)| |且 | 1使 a,b, Y a 1 . 0 X,Y Cov(X,Y) 0E(X,Y) E(X)EY)XY F(x,y) F(x)F(y) f(x)f(y)

7、E(X,Y) E(X)EY)X,YFfX,Y X,Y ( , ) ( , ; , ; )X YNX,YX,Y,Cov(X,Y) ,D(X)221212E(X) ,EY) ;D(X),DY) ;2 .2121212 X 0，E(X)2D(X)2 , 即P X | | X E(X) .22D(X)2 | X E(X)| 1, X 1即.22X ,X ,X ,12nE X D X k ( ) , ( )0,2kk nnnX E( X )X kkk当n充分大时，Y 近似 N(.k1k1i1D( X )nnkk1- ,n , p(0 p nnpn近似 (Nnnp p)统计部分设 X X ,X ,X X

8、12n1nX X ,nii111nnS (X X) (X)，nX2222n1n1iii1i11n (X X)2Sn1ii11nk, X kAknkii1） 2nXNin 2 ( ( )n X .X2222iiii1 ( ), n Y X Y (n n )2( ),则.Xn221212 设X n 则 n D X n. ( , ( ) 21n XN ( , ),则(X ) (n22ii2i1 t XX N Y ( ) n , 且X Y ( t n,tY /n yn t (n) t (n). FU n定义：设U (n ),V (n )F F(n ,n 1, 且U,V1V n12221 F (n ,n

9、 ) .12F (n ,n )21X X N(, ), X ,X ,X X 212n X,S2D(X)2 E(X) E(X) ,D(X) 2,E(S ) D(X) , 2nn nnnC X N( C , C .)22iiiiiii1i1i1X 1(n S2n2 (, )(X X)2 (n , 即X N N(2n/ ni22i1 X ,X ,XX N( , )2Y ,Y , Y12n11112n2自 2 ( , ), Y NX,YS ,S212222 22 ( ,X Y N ).12nn1212 X Y ( ) 当 t(n n 2)，22211 121212Sn nw12(n S (n S22S

10、 1122n n 2w12S /S22 ( F n2n1 /122212 f (x; , , )X p(x; , , )12k12k 1 , ,2k k1 ( , , , ) E(X)211k E(X ) ( , , , )22212k ) E(X ) ( , ,kkk12k ( , , , ) 1112k ( , , , )2212k ) ( , ,kk12k i ,i ,n (A ,A , A ).Aiiii12k( , ,)x x x12nI ) n( ;) L( ) p(x ; )nLf xiii1i1 0 u u )u )是 u( )u 是 ( ) 是 , 则E 2有效性： , 是

11、) 较D( ) D(11212 是 . , , ( , )P() 11 1 0.95.)：X 当 2 Z N(/ n( )XZ XZ , XZ .nnn222Z,)，(,X (XZ )nn , Z XXZ .nnX S/ n当 的置信区间，枢轴量t t(n 2S(X t (n,n2SSX t (n X t (nnn22SS(X t n( )，(,X ( t nnnSSX ( ，X t n( t nnn(nS2当 未知， (n2222(nS(nS(nS(nS2222(),(n(n(n(n2222(nS(nS22(0,)，(,)(n (n(nS(nS22.,(n(nS /S22当 , F(n n F2 /212 1212/122212