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p

X(x) :=P(X=x);

X i2Np

X(xi) =P(S) = 1:

??X?? EX:=X fx:pX(x)>0gxp X(x) p

X(x) =8

:12 ; x= 1 12 ; x= 0

0;??? ????? ?????? ??x?

?????EX= (1)(12 ) + (0)(12 ) =12 18 X?

EX= (0)12

+ (1)14 + (2)18 + (3)116

1 +x+x2+x3+=11x:

1 + 2x+ 3x2+=1(1x)2:

EX= 1(14

) + 2(18 ) + 3(116 14 h

1 + 2(

12 ) + 3(14 ) +i 14 1(112 )2= 1:

EX=XxP(X=x) = (3)(13

) + (4)(13 ) + (10)(13 ) =173 1 X n=11n(n+ 1)= 1: EX=1X n=1nP(X=n) =1X n=1n1n(n+ 1)=1X n=11n+ 1= +1; EX=X fx:pX(x)>0gxp

X(x) =1X

i=1x ipX(xi): EX=X !2SX(!)P(f!g): EX=1X i=1x ip(xi) =1X i=1x iP(X=xi) =1X i=1x i X !2SiP(f!g)! 1X i=1 X !2Six iP(f!g)! =1X i=1X !2SiX(!)P(f!g) X !2SX(!)P(f!g); i=1x iP(X=xi) =X !2SX(!)P(f!g) a2R? ???? ???E[X+Y] =EX+EY? ????E[aX] =aEX? f!2S:X(!) =xg \ f!2S:Y(!) =yg: ?????? ??fxigi2N??? ?????? ????X?? ??????? ??? ??fyjgj2N??? ?????? ????Y?? ??????? EZ=1X k=1z kP(Z=zk) =1X k=1 1X i=1z kP(Z=zk;X=xi)! 1X k=1 1X i=1z kP(X=xi;Y=zkxi)! 1X k=11 X i=11 X j=1z kP(X=xi;Y=zkxi;Y=yj): ???P(X=xi;Y=zkxi;Y=yj)???? ?? ?? ??????zkxi=yj? ??? ???? ????(i;j)? ???j 1 X k=1z kP(X=xi;Y=zkxi;Y=yj) 1X k=1(xi+yj)P(X=xi;Y=zkxi;Y=yj) = (xi+yj)P(X=xi;Y=yj): EZ=1X i=11 X j=1(xi+yj)P(X=xi;Y=yj) 1 X i=11 X j=1x iP(X=xi;Y=yj) +1X i=11 X j=1y jP(X=xi;Y=yj) 1X i=1x i 1X j=1P(X=xi;Y=yj)! +1X j=1y j 1X i=1P(X=xi;Y=yj)! 1X i=1x iP(X=xi) +1X j=1y jP(Y=yj) =EX+EY;

E[X+Y] =X

!2S(X(!) +Y(!))P(!) X !2S(X(!)P(!) +Y(!)P(!)) X !2SX(!)P(!) +X !2SY(!)P(!) =EX+EY: ???a2R?? ????

E[aX] =X

!2S(aX(!))P(!) =aX !2SX(!)P(!) =aEX ?P(Y= 4) =16

EX2=EY= (1)16

+ (4)16 ++ (36)16

EX2= (12)16

+ (22)16 ++ (62)16 ;P(X=1) =14 ;P(X= 1) = 38
;P(X= 2) =14 ? ???? ??Y=X2?P(Y= 1) =58 ???P(Y= 4) =38

EX2= (1)58

+ (4)38 = (1)214 + (1)238 + (2)218 + (2)214 ??? ???? ?? ???? ????EX2=P

Eg(X) =1X

i=1g(xi)P(X=xi) =1X EY=X yyP(Y=y) =X yyX fx:g(x)=ygP(X=x) X xg(x)P(X=x):

Eg(X) =1X

i=1cp(xi) =c1X

Var(X) =E(XM)2

??X EXn=X x:pX(x)>0x npX(x): ??????? ??????? ???? ? ???? ???? ??? ???X= 1?? ?? ??? ??????X=1?? ?? ??? ?????? ????EX= 0? ??XEX=X? ??? ????VarX=EX2= (1)212 + (1)212 = 1? ? ??XEX?????? 52
;32 ;12 ;12 ;32 ;52 16

VarX= (52

)216 + (32 )216 + (12 )216 + (12 )216 + (32 )216 + (52 )216 =3512

VarX=EX2(EX)2:

VarX=EX22E(XM) +E(M2)

=EX22M2+M2=EX2(EX)2: X=(

1?? ??????? ?? ????? ???

0?? ??????? ?? ????? ????

???? ?????X??? ?? ??? ?????? ??????? ????X??? ?????? ??? ????? ??[0;1)?

P(X= 0) =P((T;T;T)) =12

3=18

P(X= 1) =P((T;T;H);(T;H;T);(H;T;T)) =38

P(X= 2) =P((T;H;H);(H;H;T);(H;T;H)) =38

P(X= 3) =P((H;H;H)) =18

P(X=k) =12

nn k p

X(x) =8

:12 x= 0 12 x= 1; p

X(i) =eii!;i= 0;1;2;:::;

???? P(X= 0)?

P(X= 0) =pX(0) =e00!

=e: ???? P(X >2)?

P(X >2) = 1P(X62)

= 1P(X= 0)P(X= 1)P(X= 2) = 1pX(0)pX(1)pX(2) = 1ee2e2

E[X] =X

x:p(x)>0xp

X(x) =1X

i=1x ipX(xi):

EX= 0pX(0) + 1pX(1) =P(T):

EX= 116

+ 216 ++ 616 =16 (1 + 2 + 3 + 4 + 5 + 6) =216 =72 = 3:5: ????X??? ???? ??? ??????0;1;2;3:::????P(X= 0) =23 ?P(X= 1) =29 ;:::;P(X=n) = 23

EX= 0pX(0) + 1pX(1) + 2pX(2) +

= 023 + 123

2+ 223

3+ 323

4+ 23
2

1 + 213

+ 313

2+ 413

3+ 29

1 + 2x+ 3x2+;?????x=13

29

1(1x)2=29

113
2=22 2=12

X(4) = 3? ???X(5) =X(6) = 5?

p

X(3) =pX(5) =13

EX= 113

+ 313 + 513 =93 = 3:

EX=X(1)P(f1g) ++X(6)P(f6g) = 116

+ 116 + 316 + 316 + 516 + 116 = 3: F

X(x) :=P(X6x);

??? ???x2R?

F(x0) =X

x6x0p X(x): p

X(0) =P(X= 0) =18

p

X(1) =P(X= 1) =38

p

X(2) =P(X= 2) =38

p

X(3) =P(X= 3) =18

???? ??? ??? ???X??? ???? ??? ????? ?? ??? ???? F

X(x) =8

>>>>:01< x <0; 18

06x <1;

48

16x <2;

78

26x <3;

1 36x <1:

??limx!1F(x) = 1? ??limx!1F(x) = 0? F

X(x) =8

>>>>:0x <0; x2

06x <1;

23

16x <2;

1112

26x <3;

1 36x:

??????? P(X <3)? = limn!1FX31n =1112

10123400:20:40:60:81

????? ??? ??? ?????? ???FX(x)?? ??????? ????? ??????? P(X= 1)?

P(X= 1) =P(X61)P(X <1) =FX(1)limx!1x2

=23 12 =16 ??????? P(2< X64)?

P(2< X64) =FX(4)FX(2) =112

P(X=1) = 0:2;

P(X= 0) = 0:5;

P(X= 1) = 0:3:

???Y=X2? ???E[Y]? ????pY(1) = 0:2 + 0:3 = 0:5???pY(0) = 0:5? ?????E[Y] = 00:5 + 10:5 = 0:5?quotesdbs_dbs19.pdfusesText_25