, , . . . , . , , 0 . , . , , .
, n w1,,wn, , m wi1,,wim, , , , .. P(A)=m/n.
l L. L . , l L, l : P= l/ L. g G. G . , g G, g, g : P= g/ G. v, V: P= v/ V.
, , , . . , , , , , , : (), (), , . , ..
A
B = B
A A
B = B
A
, S : 1. A . S ()>=0. 2. ()=1. 3. : . S, . S, *=0, (+)=()+(). -: , ={wi1,,wim},, , ()= l 1 m il, 0<=pil<=1, l=1,,m, ()>=0, .. 1 . 2 , ={w1,,wn} , ()= l 1 m il, ()= i 1 n pi=1. 3 , . , , .
: (+)=()+()-(). . , (++)=()+()+()-()-()-()+(). ={wi1,,wim} ={wj1,,wjk} , =+ .
1) (/B)=P(A*B)/P(B), ()>0. 2) , (/B)=P(A). , .. , . ()>0 (B/A)=P(A*B)/P(A)=P(A/B)*P(B)/P(A)=P(A)*P(B)/P(A)=P(B). : (*)=()*(/A). : (*)=()*(). 3) 1,2,,n , , .. i*Aj=0, i =j, U i 1 n i=.
, , : ()=()*(). ()=()*(), .. .
, () 1,2,,n , , : ()=(1)*b1(A)+P(B2)*Pb2(A)++P(Bn)=1. 1,,n, P(Ai)>0 , : ()= i 1 n P(Ai)*P(B/Ai).
: ()>0 i, P(Ai)>0, B . U i 1 n i . (k/B)=P(Ak)*P(B/Ak)/ i 1 n P(Ai)*P(B/Ai).
=(w), , w . . .. , , , , .. , 1, 2, 3, , P{X=xn}=pn>=0, n=1,2,3 1+2+3+=1. ... , n n=P{X=xn} , 1+2+3+=1. , ... , -, ..
Fx(x) C.. Fx(x)=P{w:X(w)<x}. Fx(x)=P{X<x}. , , F(x) Fx(x). : 1) .. [0,1]: 0<=F(x)<=1. , .. - {X<x}, - . 2) - , .. [x1,x2), F(x2) F(x1): {x1<=X<=x2}= F(x2) F(x1). 3) .. , .. F(x2)>=F(x1), 2>1. 4) {X>=x}=1 F(x). 5) , F(x) 1. 6) , F(x) . 7) .. , .. lim . +0 F(x )=F(x).
. ... : ()=11+22++npn. ... , ()= i 1 xipi, . , . . : 1) . : ()=. 2) . : ()= (). 3) . .. . : (1,2n)=M(X1)*M(X2)M(Xn). 4) . .. . : (1+2+3++n)=M(X1)+M(X2)+M(X3)++M(Xn).
.. . .. . : D (X)=M[X M(X)]*2. : D (X)=M (X*2) [M (X)]*2. : 1) . : D(C)=0. 2) ., : D (CX)=C*2D(X). 3) . .. . : D (X1+X2++Xn) =D(X1)+D(X2)++D(Xn).
... - n , , - =k ( k ) : Pn (k)=Cn*k p*k q*n-k.
, , : Pn (k)=*k e*-/k!, n , =np ( n ), , .. .
.. , Pm=P{X=m}=q*m p, m=0,1,2,, 0<p<1, q=1-p. , , m 0 Pm=1: m 0 Pm= m 0 pq*m=p*1/1-q=1. .. , .
.. , () , Fx(x) : Fx(x)= px(y)dy. .., () , , , . .. : f(x)=F(x). , ... , (,b), P(a<X<b)= b f(x)dx. F(x)= f(x)dx. : 1) .. , .. f(x)>=0. 2) : f(x)dx=1.
. ... , , : ()= f(x)dx, f(x) - .. . , . , (,b), ()= b xf(x)dx. . , , ... ...
... , , : D(X)= [x-M(X)]*2f(x)dx, : D(X)= x*2f(x)dx [M(X)]*2. , (a,b), D(X)= b [x M(X)]*2f(x)dx, D(X)= a b x*2f(x)dx [M(X)]*2. ... ...
... , (,b), , , f(x)=1/(b-a); f(x)=0. , ()dx=1. .., , , .. (,+) . [a,b], : P{x<X<x+}= + 1/b-adt=/b-a. : F(x)=0, <=a, x-a/b-a, a<x<=b,1 >b.
... , , , .. x>=0 ()=* - * x<0 ()=0. .. F(x)= (t)dt=0, x<=0,1- * x>0.
... >0, : ()=1/( 2 *) * 1/2*(x-a/)*2. , . N(a,). ()=1/( 2)* *2/2 =0 =1, ()=1/( 2)* t*2/2dt, P{a<=n-np/( npq)<=b}, =0, =1.
,
f(x)=(1/s Ö (2p ))*e-(x-a)2/2s 2; s >0.
(Z=x-a/s )
()=
.
.
-;
(2)=
> (2)>(1)
3.(+¥ )=0,5..
(¥
)=
- () 0,5.
.
x . s (s >0).
.., . , . . , , , .. t>0. P{|X MX|>=t}<=1/t*2 M(X MX)*2=1/t*2 DX . .., ; .. .
. .. 1,2,3,,Xn, . ( ), .. . , .. , : lim n P(|1/n i 1 n Xi 1/n i 1 n M(Xi)|<)=1. , , . , . , .. , : lim n P(|1/n i 1 n Xi a|<)=1. . P{|X|>=t}<=1/tM|X| - . -: 1) ... . ..., {X=xi}=pi, i=1,2,3,, i 1 pi=1. {|X|>=t} pi, xi (-t,t). , xi, (-t,t), |xi|/t>=1. : P{|X|>=t}= i: |xi|>=t pi <= i:|xi|>=t |xi|/t pi<= i:|xi|>=t |xi|/t pi+ i:|xi|<t |xi|/t*pi =1/t i 1 |xi|*pi=1/t*M|X|. 2) ... . ... (). , |X|>=t, (-, -t) (t,). |x|/t*t>=1. |x|/t*p(x)>=0, t t |x|/t*p(x)dx>=0. M|X|= |x| p(x) dx, .
. , n , (0<p<1), k (, ), ( , n). Pn(k)=1/( npq)*(). ()=1/( 2)* *2/2, x=k np/( npq). . , n , (0<p<1), k1 k2 , : P(k1;k2)=() (). ()=1/( 2) * 0 (z*2/2)dz , =(k1 np)/( npq), =(k2 np)/( npq).
.. (,Y), (x,y). Y, , .. , . , . ... . ... F(X,Y), (,y) , , , Y , y: F(x,y)=P(X<x,Y<y). :1) : 0<=F(x,y)<=1. 2) :F(x2,y)>=F(x1,y), 2>x1. F(x,y2)>=F(x,y1), y2>y1. 3) : 1) F(-, )=0, 2) F(x,-)=0, 3) F(-, -)=0, 4) F(, )=1. 4) ) = : F(x,)=F1(x). ) = : F(, )=F2(y).
. 1) .. X Y : x1,x2,,xn; y1,y2,,ym. Y=yj (j ) p(x1|yj), p(x2|yj),,p(xn|yj). Y. Y : p(xj|yi)=p(xi,yj)/p(yj), p(yj|xi)=p(xi,yj)/p(xi).
x h . . m x h =((x (x ))*(h (h )))
:
m x h =(x *h )(x )*(h ) : m x h =((x (x ))*(h (h ))) .
m x h =(x h (h )h (x )+(x )*(h ))=(x h )(h )*(x )(x )*(h )+(x )*(h )=(x h )(x )*(h )
, x h , m x h =(x h )(x )*(h )=(x )*(h )(x )*(h )=0; m x h =0. , =0, , . m x h 0, x h . x h , 0 0. , x h . . x h x h , :
x h =m x h /s (x )*s (h ) x h x h , x h . (x ,h ) .
x h =± 1, x h .
x h <0, , .
.
D(x ± h )=M((x ± h )2)M2(x ± h )=M(x 2± 2x h +h 2)(M(x )± M(h ))2=M(x 2)± 2M(x h )+M(h 2)+M2(x )+2M(x )*M(h )M2(h )=D(x )+D(h )± 2(M(x h ))M(x )*M(h )=D(x )+D(h )± 2m x h
. , . {,S,P} (.. S ) .. . n 1,2,,Xn, n .., .. . , n .. , .
.. .. 1,2,,xn , .. . , .. F(x) .., , . F^n(x) .. F(x), xi , .. X*1=minXi, ..; , . *1<=X*2<=X*n, , . . , , h, ni/h ( ), ni i- .
K * K f(X1,X2,,Xn) .. X1,X2,,Xn. , K *=f(x1,x2,,xn), 1,2,,xn n (). , . . , . . (. ) : =( i 1 k nixi)/n, xi , ni xi, n= i 1 k ni . : D=( i 1 k ni(i-X)*2)/n. : s*2=n/n-1*D= ni(xj X)*2/n-1. . , . , : v1=M1. , v1=M(X) 1=, ()=. , . , v1=M(X),M1=,=D(X),m2=D, : ()=, D(X)=D.
. ... ..., n 1,2,,xn. , , K , ; K *=K (x1,x2,,xn). , xi (xi;K ). ... K : L (x1,x2,,xn;K )=p(x1;K )*p(x2;K )p(xn;K ). K K *, . L lnL K , L , , lnL. ... ..., n 1,2,,xn. , f(x) , K , . ... K : L(x1,x2,,xn;K )=f(x1;K )*f(x2;K )f(xn;K ).
, , . , . 1. . : t(/ n)<a<+t(/ n), t(/ n)= , n , t (t), (t)=/2; ( n<30) t (s/ n)<a<+t (s/ n), s- . 2. ( ) s s(1-q)<<s(1+q), q<1; 0<<s(1+q), q>1. 3. ( ) w ( 1 2).
K ^=X=1/n i 1 n Xi . =K f(x,K )=1/( 2 )* (-K )*2/(2 ) n. i Dxi= , . , x , s 2. n x 1, x 2, ... , x n. . M |x | > a j. :
; , x
1, x 2, ... , x
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.
M(x i) = a; (x
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:
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(H0). : x , , H0: M(x ) = a. , : H1: M(x ) ¹ a;
H1: M(x ) > a, H1: M(x ) = a1. , , k , .
, 0 , k. 0 , k.Î (k. .; k. .) k. ( k Î (- ¥ ; k. .) È (k. . ; ¥ ), 0 1. : : 0 , , . : P(H1/H0) = a - . , a . : 1, , . b = P(H0/H1) (1-b ) - - , .
1-b = P(H1/H1)
x h .
x 1, x
2, ... , x n n, h
1, h 2, ... , h
m m, (
;
).
: H0: M(x ) = M(h
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(
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=
M(Z) = 0; (Z) = 1. , Z. a , , .
P(0 < Z < Z.) + P(Z > Z. .) = ½ (Z.) + a /2 = ½ Ô(Z. .) = ½ - a /2
Z. =
|Z.| < Z.. Þ 0 |Z.| > Z.. Þ 0 .
x h , , , . x 1, x 2, ... , x n h 1, h 2, ... , h m
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:
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k = n + m 2 ( k a , t., t.).
. . = - . . | . | < . . Þ 0 | . | > . . Þ 0 .
, S. ., - - . , , - . - . - , . t: - t . t (t=1,2,3,...), ( ), t , . , , , . , t. S1, S2,...,Sn -. , . -, - t. 1(t)- , t - 1- . , , . , Xi=Si ti , Xi-1=Si, ti-1. , . , , , - - , - , . 1 . . .
. i(t)- : i=1,n( ), i(t)
d i(t)/dt=å ( i<>k,k=1 n) l ki* i(t)å ( j<>1,j=i n) l ij*Pi(t); i=1,n( ) (1) n , .. t å ( i=1 n) Pi(t), (1) - . , t=t0, P1(t0)=1, Pi(t0)=0, i=1,n( ).
- - . - - Pi(t), i=1,n( ).
, , ,
, i(t); i =
, ,
..
= i.
, , , .
, . , t , Si Sj.
, t ® ¥ Pi=const.
, Þ ,
, n , 1, 2, ... , n.
, , Si Si+1 Si Si-1 .
l i, i+1, l i, i-1 -
.
:
S0: l 01P0 = l 10P1 S1: l 10P1 + l 12P1 = l 01P0 + l 21P2 S2: l 21P2 + l 23P2 = l 12P1 + l 32P3 ... Sn: l n, n-1 Pn = l n-1, n Pn-1 P0 + P1 + P2 + ... + Pn = 1
P1 =
l 01P0 + l
12P1 =
l
01P0 + l 21P2
P2 =
P3 =
Pn =
...
P0 +
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(a > 0)
a = t l , l -
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t, t ³ 0
F(t) = l e-l t
, ().
;
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=
Þ l =
Η , - . . , , - ( , . 1- ). n l . , =1/m ; . . , m . m (., . - ).
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:
0=(1+
)-1
P1=
; P2=(l
2/(2!m 2))*P0;....;r=(l
k/k!m k)*P0
:
Η , - . . , , - ( , . 1- ). - , - l , m . - m . , , , . , , , , . , , , . : S0 S1 , n-1 , Snn , , Sn+1 , 1 , m-1 Sn+m , m () .
:
0=(1+
1.=n+m=
2. - q=1Pn+m 3. - A=l q 4.
Η , - . .
, , - ( , . 1- ).
. - , , m>¥ . 0
0=
=
m >¥ å 1 + e + e 2+ ...+e m-1 , 0<e <1; e >=1 , .. . , - .
, m>¥ , e <1 . . - - . - .:
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