.
2 :
1) : , .
1. : R U. ={ I=U/R}.
2) : . .
2. . A={ }.
.
.
: , mn(A) n P*(A). mn(A)/n= P*(A), n .
P*(A) . , ..
2 . .
.
. ().
- ( ) , .
- - . . , .
:
1) - . .
2) Ǿ . .
3) + - . .. + .
4) . - . .. .
5) , - \. .. , .
6) , (). .. , , .
7) , ,
8) .
-:
1):
+=+; =.
2):
(+)+=+(+); ()=().
3):
(+)=+; +=(+)(+).
3 .
, , , .
:
) .
) .
|
|
.
, - , - . :
()= / - .
-:
1)()>0
2)
3) , (= Ǿ), (+)=()+().
4
R1 R2 n Rn.
- , , R3-, Rn- .
, , .
(mes A) . .
, - -. P(A)=mes(A)/mes()
1. 2 . . 8 . ?
-
x |
y |
(,) R2
={(,) | }
A = {(,) | |x-y| 1/3}
mes()=1, mes(A)=5/9;
P(A)=5/9
C-:
1)()
2)
3) .
5
, P(A). , - F P(A), .
+, , -, , , , Ǿ, F, .. F .
-, F - - .
1. ={ 4 1,2,3 4 }
F:
Ǿ
- . , F . (), F, , :
1) () , ( )
2) ( )
3) , = Ǿ. (+)=()+() ( )
1)
2) -
1
6 -
- :
1) ()
2)
3)= Ǿ => (+)=()+()
-:
4)
5) (Ǿ)=0
6)
7)
8)(+)=()+()-()
7 . .
. (/).
=> N(B) . N(AB) A
. (,F,P) . , , , . - :
. 1), :
2) - - .- .
1.
2.
3.
4) - - -.
5) . - P(A/B) ., , - .
6) n :
8 , . .
. , P(A/B)=P(A)
|
|
:
1) , .. P(B/A)=P(B)
.. .
2) , P(AB)=P(A)*P(B) :
. 1,A2,A3,,An , n .
. A1,A2,,An : P(A1,A2,,An)=P(A1)*P(A2)P(An)
.
9 .
H1,,Hn . H1+H2++Hn=
{Hz} (Hi,Hj 0, i j), , .. .
.
,H1,..,Hn, {Hi} ,
A=A* =A(H1++Hn)=AH1++AHn
P(A)=P(AH1)+P(AH2)++P(AHn)=> .
P(A)=P(H1)P(A/H1)++P(Hn)P(A/Hn)
10
H1,,Hn . H1+H2++Hn=
{Hz} (Hi,Hj 0, i j), , .. .
P(Hk/A)=(P(Hk)P(A/Hk))/P(A)
P(A)*P(Hk/A)=P(A*Hk)=P(Hk)P(A/Hk) /: P(A)
P(Hk/A)=(P(Hk)P(A/Hk)/P(A))
11
n n .
n ,
1) z (-, () )
2) , .. P(A) .
3) p q=1-p
- pn,m m .
- :
= ,
. (,F,P) . , . X=X(w), - x () - x = { } F. - -, - , x
. (-) x : Fx(x)=P{X<x}
- Fx(x)
1 P{a<=x<b}=Fx(b)-Fx(a)
{x<b},{x<a},{a<=x<=b}
{x<b}={x<a}+{a<=x<=b}
2 P{a<=x<=b}=Fx(b+0)-Fx(a)
3 P{a<x<b}= Fx(b)-Fx(a+0)
4 P{a<x<=b}=Fx(b+0)-Fx(a)
5 P{x=a} = Fx(a+0)-Fx(a)
1 Fx(x)
2 0<=Fx(x)<=1
3 Fx(- )=0, Fx(+ )=1
4 Fx(x) t a R
13
X, - ()
Fx,
Fx(x)=P{X<x}=
=1
- Fx(x) :
)
) Fx(x)=0 x<x1
) xi 1-
14
X n, p(X~B(n,p)), X 0,1,,n - p(n,k)= P{X=k}=
B(n,p) n - p.
. X-C - -, m=M[x]= - (m-..X)
|
|
-:
X= , Xk 0 1
P q p
M[x]=
B(n,p):
.. Xk pq,
D[X]= npq
15
n-> p->0 np= =const,
X 0,1,2,,k - pk=p{X=k}= , >0 - (X~Pn())
- Pn() - (- )
:
: Dx=
.
. . .
. X - , Fx(x)( -), :
Fx(x)=P{X<x}=
- fx(x):
1. P{a<=X<b}=
2. a IR P{X=a}=0
3. fx(x)>=0
4. (
5. -: fx(x)=Fx(x)
17
t .
X - - m,t(X~N(m,t))
N(0,1) :
(x)= -
- (-x)=(-(x)), x>=0 (x) x>=0
M[x]= -> M[x]=m
D[x]=
x~N(m, ) P{a<x<b}
P{a<x<b}=
P{/X-m/<l}=2(l/ )-1
18. . , .
19. . . . .
21. , .
22.
. k- X
α k [X]=M[Xk]
1) α 1 [X]=M[X]
2) X => α k [X]=∑ Xkp
. . k- X
Μk[X]=M[(X-M[X])k]
1) . X-M[X]=X ( ) - . . .
2) μ1[X]=0
α k [X] μ k [X].
μ k =M[(X-M[X]) ]= M[ X (-1) (M[X]) ]=
= M[X ](M[X])
=> μ k [X] = α j [X] * α [X]
23.
. . X . μ 2 [X]:
D[X] = M[(X-M[X]) ]
X : D[X] = p i
D[X] , X M[X].
. X . T[X] =
:
1. D[X] , 0
2. D[C] = 0, C=const
3. D[X] = M[X ]-M [X]
4.D[cX] = c D[X]
5. X Y , D[X+Y] = D[X]+D[Y]
D[X+Y] = M[(X+Y-M[X+Y]) ] = M[(X-M[X]+Y-M[Y]) ] =
= M[(X-M[X]) ]+M[(Y-M[Y]) ] + 2M[(X-M[X])]*M[(Y-M[Y])] = D[X] + D[Y]
24. .
. X f x (x), f x (x)dx , . X M[X] = f x (x)dx
. X f x (x), f x (x)dx , X : D[X] = f x (x)dx
.
1)M[X] X , X-
2)- . . . . . . M[X].
1. X~N(m,τ);M[X] -?
M[X] = dx== m = M[X]
2. X~N(m,τ);D[X] -?
|
|
D[X] = = dx= =
25. .
26. : , , , .
27. , .