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http://www.Beiki.info X M M 2 tx e 3 ( tx ) ( tx ) 1 tx ... 2! 3! 2 tx o tx E (e ) x e f ( x) x 3 ( tx

) ( tx ) [1 tx 2! 2 3! ...] f ( x) 3 x f ( x) t x xf ( x) t x x f ( x) t x x f ( x) ... 2! 3! 2 2 3 3 r 1 t t t ... t ... 1 2! 2 3! 3 r! r http://www.Beiki.info 6 o d MX(t) : E ( X ) M (t ) ] r :2 dt 3

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n p<=1=<0 X . 8 http://www.Beiki.info o x n x n tx tx n : M X (t ) E (e ) x0 e C x p (1 p) n t n x 0 C x ( p e ) M (1) X t x n x (1 p) t (t ) np e [ p e t ( 2) t

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:43 X x ; x 0,1,2,3,... !x . tx X tx e !M X (t ) E (e ) x0 e x : x t ) ( e t t e x 0 ])e Exp( e ) Exp[ (e 1 !x http://www.Beiki.info 10 x e f ( x) o : X Y ) MX(t ) MY(t t ) MX(t)=MY(t X Y . 11

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Z Z=X+Y Z 14 http://www.Beiki.info o : x y x y x y g ( x) e dy e ;0 x , h( y ) e dx e ;0 y 0 M 0 X tx (t ) e 0 e x ty y 1 1 dx ; t 1, M Y (t ) e e dy ;t 1 0 1 t 1 t 2 1

M (t ) M (t )*M (t ) ( ) ; t 1 1 t Z X Y http://www.Beiki.info 15 o : ) f(x,y ( E ) M Y (t1 ,X et X t Y )t 2 X ,Y X Y . : M X ,Y (0, t 2) M Y (t 2), M X ,Y (t1 ,0) M X (t1), M X ,Y (0,0) 1 2 1 k m ) m Y k ) ( , E ( X X ,Y t1 t 2 ] 0 , 0 t1 t 2 M t t m

k 2 1 2 ] )(t1 , t 2 t10,t 20 16 X ,Y M t t 2 k1,m1 E ( XY ) 1 http://www.Beiki.info o Y X : M X ,Y (t1 , t 2) M X (t1) * M Y (t 2) f ( x, y ) g ( x ) * h( y ) t1 X t 2Y ) t1x t 2 y f ( x, y )dxdy ( , ) E ( M X ,Y t1 t 2 e e e: X Y x y

et1 g ( x)dx et 2 h( y )dy M X (t1) M Y (t 2) X y t1 X t 2Y ) E ( t1 X ) E ( t 2Y ) ( , ) ( ) ( ) E ( e e e e M X ,Y t1 t 2 M X t1 M Y t 2 t1xt 2 y f ( x, y )dxdy t1x g ( x)dx t 2 y h( y )dy t1xt 2 y g ( x)h( y )dxdy e e e e X Y X y X Y f ( x, y ) g ( x ) h ( y ) http://www.Beiki.info 17 o : X ) f(x :

Xx X )(t ) E (t ) x t f ( x X ||t d . (1) k 0 xf )( x E ( X ) <=1 X dt : 18 http://www.Beiki.info o :48 X X

x (t ) E (t ) x t f ( x) d (1) k 0 xf ( x) E ( X ) X dt http://www.Beiki.info 19 o :48 X x n x n f ( x) C x p (1 p) ; x 0,1,2,3,..., n n p<=1=<0 x x n x nX x n )(1 p X (t ) E (t ) x0 t C x p . x n x n n n : ] x 0 C x ( pt ) (1 p) [ pt 1 p )(1 , X (1) np 1 20 n 1

]p (t ) np[ pt 1 http://www.Beiki.info )(1 X o :1 a b X : E(aX+b)=aE(X)+b : E (aX b) (ax b) f ( x)dx a xf ( x)dx b f ( x)dx aE ( X ) b E(b)=b :1 E(aX)=aE(X) :2 21 http://www.Beiki.info o :2 /

X / : E ( g ( X ) h( X )) ( g ( x) h( x)) f ( x)dx : ]) g ( x) f ( x)dx h( x) f ( x)dx E[ g ( X )] E[h( X 22 http://www.Beiki.info o :2 / X Y / : E[u ( X , Y ) v( X , Y )] [u ( x, y ) v( x, y )] f ( x, y )dxdy : u ( x, y ) f ( x, y )dxdy v( x, y ) f ( x, y )dxdy ]) E[u ( X , Y )] E[v( X , Y ]) E[ g ( X ) h(Y )] E[ g ( X )] E[h(Y :1 :2 23 ) E[ X Y ] E ( X ) E (Y http://www.Beiki.info

o Y X :4 : E(XY)=E(x)*E(Y) f ( x , y ) g ( x ) h ( y ) E ( XY ) xyf ( x, y )dxdy : E ( XY ) xyg ( x)h( y )dxdy xg ( x)dx yh( y )dy E ( X ) * E (Y ) http://www.Beiki.info 24 2 aX b 2 aX b o

X b a :1 2 2 2 2 : aX b Var (aX b) a Var ( X ) a X E{[(aX b) 2 aX b] }, a b : E (aX b a b) a E ( X ) a 2 X X aX b 2 X 2 2 2 X 2 2 a :1 2 X b aX 2 X 2 X

:2 http://www.Beiki.info 25 Y X :2 f(x,y) 2 2 2 2 aX bY Var (aX bY ) a Var ( X ) b Var (Y ) 2abCov( X , Y ) a 2 aX bY E{[(aX bY ) 2 b2 2 X Y 2ab XY aX bY ] }, a b : E{[(aX bY ) (a X b Y )] } E{[(aX a X ) (bY b Y )] } a E[( X ) ] b E[(Y ) ] 2abE[( X )(Y )] a 2 b2 2ab X Y 2 aX bY X Y 2 2

o 2 aX b 2 2 2 2 X 2 aX bY a 2 X b2 2 Y 2 2 X Y XY 2 Y Y X :1 2 Xn2 ...2 2 X2 X21 :2 2 2 aX 1 1

a2 X 2 ...an X n a1 X1 a 2 http://www.Beiki.info X2 ... a n Xn 26 o :3 X Y 1 1 1 . XY 2(1 ) : 1 XY 1 27 XY XY ) XY ) 2Cov( X , Y Y

2(1 X Y ) 2X Y2 X ) 2Cov( X , Y Y 2 2 X ( 0 Var X Y ( 0 Var ) Y X 2 2 Y 2

X 2 Y X Y Y X X http://www.Beiki.info o Cov(X,Y)=Cov(Y,X):1 Cov(X,X)=Var(X) :2 Cov(a1+c1X,a2+c2Y)=c1c2Cov(X,Y) :3 Cov (a1 c1 X , a2 c2 Y ) E[(a1 c1 X [c1 E ( X : a c )(a c Y a c 1 1 X 2 2 2 2 Y )]

) c E (Y )] c c Cov( X , Y ) X 2 Y 1 2 http://www.Beiki.info 28 o Z Y X :4 fX,Y,Z(x,y,z) Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z) Cov( X , Y Z ) E[ X (Y Z )] E ( X ) E (Y Z: ) E ( XY ) E ( XZ ) E ( X ) E (Y ) E ( X ) E ( Z ) [ E ( XY ) E ( X ) E (Y )] [ E ( XZ ) E ( X ) E ( Z )] Cov( X , Y ) Cov( X , Z ) http://www.Beiki.info 29 m o n Y X :5 ... X1 Xm2 n

n m Cov(i 1 X i, : ) Cov( X ) Y1 Y2 Xn Ymi,Y ... j j j 1Y i 1 j 1 j 1 j i 1 i n n : m m E ( X i ) , E (Y j ) j E (i 1 X i ) i 1 , E ( j 1Y j ) j 1 j i i n m n n Cov(i 1 X i, j 1Y j ) E[(i 1 X i m

m )( Y )] E[ ( X ) (Y )] E[ ( X )(Y )] E[( X )(Y )] Cov( X ,Y ) n i 1 m i i 1 n m i 1 j 1 i i j 1 i j j j j http://www.Beiki.info n m i 1 j 1 n

m i 1 j 1 j j 1 i i j 1 j i i j j j 30 n n n n m o : | X j Y j Var (i 1 X i ) Cov(i 1 X i, j 1 X j ) i 1 j 1 Cov( X i , X j ) , ) 2 i j Cov ( , )

i j Cov ( m j 1Var ( X i ) i j Cov( X i , X j ) Xi Xj X iXj n m Var (i 1 X i ) j 1Var ( X i ) 2i j Cov( X i , X j ) : Xj Xi n m Var (i 1 X i ) j 1Var ( X i ) http://www.Beiki.info 31 o :59 ... X1 X2 Xn 2 n 1 X i 1 X i n S2 2 n S 2

) i 1 ( X i X 2 S 2 ] S n 1 [Var ( X ), E n 1 32 . http://www.Beiki.info 2 1 1 1 n n Var ( X ) Var ( i 1 X i ) ( ) Var (i 1 X i ) ( ) n .: 2 n Var ( X i ) n 2

2 n 2 n n n S ( X i X ) [( X i ) ( X )] ( X i ) ( X ) 2( X ) ( X ) ( X i ) n ( X ) 2( X )n( X ) ( X i ) n ( X ) E ( S ) E[( X i ) ] n ( X ) n nVar ( X ) (n 1) E ( S ) 2 n 2 i 1 2 n i 1 2 n n i 1 i 1 2 2 n i n

i 1 2 2 i 1 2 2 n 2 i 1 2 n i 1 o i 1 2 2 2 2 n 1 http://www.Beiki.info 33 o :1 X a ) t ( X a at

tX at atM ( t ) E ( ) E ( ) M (t)=e )(t X+a e ) e eX e M X (t M X a : :2 X a ) t ( aX ( at ) X ( t ) E ( ) E ( e e ) M X (at ) : M aX )MaX (t)=MX(at : 34 http://www.Beiki.info o

:3 ... X1 X2 Xn (t ), i 1,2,..., n M X n i 1 a i Xi Y n ( t ) ) (ai t X: M Y i1 M i i tY ) t ( a1 X 1 a2 X 2 ..an X n t ( a1 X 1 t a2 X 2 t an X n ( t ) E ( ) E ( ) E ( ... e e e e : )e MY ta X i i ) M ( ai t a1 X 1) E ( a2 X 2)...E ( an X n) e )

X i e e ) ( ai t 35 n Xi (an t ) i 1 M Xn (E t t (a2 t )... M (a1 t ) M X2 X1 (t E (e M Y (t ) M http://www.Beiki.info o :1 . :2 X Y X ) t( (t ) * Y ) t( Z=X+YZ

Y : )t X ) E (t X Y Z (t ) E (t ) E (t Y Z X ) E (t ) E (t ) X (t ) * Y (t 36 http://www.Beiki.info o RY [ ]X,Y Y=y | E ( X y ) x f Y=y ( x | y X ) X |Y X |Y R 2. 2

) Var ( X | y ) X |Y ( x X |Y ) f ( x | y X |Y R y y o ( x | y )dx ( x | y )dx 37 X |Y f 2 X |Y f x Ry X |Y E ( X | y ) ) Var ( X | y ) X |Y ( x X |Y Ry http://www.Beiki.info 2 o Var ( X | Y y ) E ( X 2 | Y y ) [ E ( X | Y y )]2 E (u ( X | y ) u ( x) f X |Y ( x | y )dx RY n n

] E (i 1 X i | Y y ) i 1 E[ X i | Y y 38 http://www.Beiki.info :65 E(X|Y=y) . y Y X e e f ( x, y ) ;0 x ,0 y . x x y y y x h( y ) ( y )e e dx e [ e y ]0 e y 1 y y 0 x y y x ( ) e e f ( x, y ) y y 1 f X |Y ( x | y )

( ) e y y h( y ) e 1 o x x o x E ( X | Y y ) ( )e dx [ xe ye y ]0 0 y lim x xe x y x y y y y http://www.Beiki.info 39 o

: X Y ) Y(X|Y)[=E(X . ) y E ( X | Y y)h( y) E ( X ) E ( X | Y y)h( y)dy E ( X : |EY]EX ) E ( X | Y y )h( y ) y [ x xf ( X x | Y y )]h( y ) f ( X x, Y y ) ]h( y ) x x y f ( X x, Y y ) h( y y [ x x ) xg ( x) E ( X 40 y http://www.Beiki.info x o :68 3 3 5 7 . . )E ( X ) y E ( X | Y y )h( y ) E ( X | Y 1) P (Y 1) E ( X | Y 2) P(Y 2) E ( X | Y 3) P(Y 3 : X Y 1 ) [ E ( X | Y 1) E ( X | Y 2) E ( X | Y 3)]; E ( X | Y 1) 3; E ( X | Y 2) 5 E ( X 3 1

; E ( X | Y 3) 7 E ( X ) E ( X ) [3 5 E ( X ) 7 E ( X )] E ( X ) 15 3 41 http://www.Beiki.info o Y X : : Var ( X | Y ) E ( X 2 | Y ) [ E ( X | Y )]2 Y )]E ( X ) E[Var ( X Var(E(X|Y)[+E(Var(X|Y)[=Var(X) | Y )] E[ E ( X 2 | Y )] E{[ E ( X | Y )]2 } E[ E (X | 2 2 : E[Var ( X | Y )] E ( X ) E{[ E ( X | Y )] } 2 2 )]E ( X ) Var [ E ( X | Y )] E{[ E ( X | Y )]2 } {E[ E ( X | Y )]}2 E[ E (X |Y Var [ E ( X | Y )] E{[ E ( X | Y )]2 } [ E ( X )]2 E[Var ( X | Y )] Var [ E ( X | Y )] E ( X 2 ) [ E ( X )]2 Var ( X ) http://www.Beiki.info 42 o :70 ... X1 X2 N N ) Var (i 1 X i N N E[i 1 X i | N .

] NE ( X );Var (X ) [i 1 X i | N ] NVar N N N : ]) Var ( X i ) E[Var ( X i | N )] Var [ E ( X i |N i 1 i 1 i 1 ]) E ( NVar ( X )] Var [ NE ( X 43 ) E ( N )Var ( X ) [ E ( X )]2 Var ( N http://www.Beiki.info o F F X=1 F X=0 )E(X)=P(F Y ) P( F ) E ( X ) y P( F | Y y )h( y E(X|Y=y)=P(F|Y=y);P(F)=E(X)=E [)Y]EX|Y(X|Y P( F ) E ( X ) P( F | Y y )h( y )dy 44 http://www.Beiki.info

o Y X :73 h(y) g(x) . Z=X+Y FZ ( z ) P ( X Y z ) P( X Y z | Y y )h( y )dy: P( X z y )h( y )dy G X ( z y )h( y )dy d d f Z ( z ) FZ ( z ) G X ( z y )h( y )dy dz dz d dz GX ( z y)h( y)dy g X ( z y)h( y)dy http://www.Beiki.info 45 o :74 X Y f X ( x) fY ( y ) 1;0 x 1,0 y 1 Z=X+Y . 1 f X Y ( z ) g X ( z y )(1)dy : 0 0 y 1 1 y 0 z 1 z y z z 0 z 1 f Z ( z ) (1)(1)dy z 0

1 46 1 z 2 f Z ( z ) (1)(1)dy 2 z z 1 http://www.Beiki.info o Y X : : X g(X) E{]Y-g(X)[2<=E{](Y-E(Y|X)[2} E{[Y g ( X )]2 | X } E{[Y E (Y | X ) E (Y | X ) g ( X )]2 | X } : E{[Y E (Y | X )]2 | X } E{[ E (Y | X ) g ( X )]2 | X } 2 E{[Y E (Y | X )][E (Y | X ) g ( X )] | X } E{[Y E (Y | X )][E (Y | X ) g ( X )] | X } [ E (Y | X ) g ( X )]E{[Y E (Y | X )] | X } [ E (Y | X ) g ( X )][E (Y | X ) E (Y | X )] 0 E{[Y g ( X )]2 | X } E{[Y E (Y | X )]2 | X } E{[ E (Y | X ) g ( X )]2 | X } 2 g ( X )] | X }0 E{[ E(Y | X) E{[Y g ( X )]2 } E{[Y E (Y | X )]2 } http://www.Beiki.info 47 o :75 X Y )E(Y|X . a b Y X : a X g(X)=a+bX , b Y X X :

Y X a b 2 2 ]) E{[Y (a bX } E[Y 2 2aY 2bXY [) a 2 2abXE{]Y-(a+bX }( b 2 X 2 ]; E (Y 2 ) 2aE2 Y ) 2bE ( XY ) a ) 2abE( X ) b 2 E ( X 2 2 ) E{[Y (a bX )]2 } 2 Y 2a 2b X 0; E{[Y (a bX )]2 } 2 E ( XY 2a X 2bE ( X ) 0 a b Y X , b Y X X 48 http://www.Beiki.info a Y

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