.
䳿 n , n m , m , :
,
℮
.
.
_________________________________
.
䳿 n , n 䳿 m mj :
,
,
, .
___________________________________
18. .
,
,
W() p 䳿 . p 䳿 W() 䳿 n .
䳿 ôW(A) ô< e (e > 0 ). n , :
.
_________________________________
.
np=a= =const 䳿 m :
,
.
n (m) m = np.
_________________________________
20. . .
, 䳿 Ώ , Ώ , 䳿 R1 n- Rn.
. , Ώ R1, . Rn, n-.
, . . .
, , .
, , .
_________________________________
21. .
|
|
, , , : F(), .
, 䳿 < x, :
F(x) = P(X < x) (1)
F(x):
1.
2. , , .
_________________________________
22. ٳ .
, f (x). ٳ F(x):
f (x)
1. .
2. :
3.
4.
_________________________________
23. .
.
, Ω,
Ω ,
Î [; b], () Î [; b], : [; b], .
_________________________________
24. .
1. :
() = .
2. () = ().
.
:
3. ,
.
_________________________________
25. .
(o) , .
, :
f (M) = max.
, ; . . , . .
() , :
, :
(2)
Î [; b]:
. (3)
, , , , = , , f (x), .
_________________________________
26. .
. , .
( ())
|
|
.
;
.
Î [; b],
.
, , .
, . .
:
.
_________________________________
27. .
1. ,
.
.
2. . (2)
:
3. ,
.
:
;
䒺 .
.
_________________________________
28. .
k- k:
.
;
.
k- ( ())k:
(5)
.
_________________________________
29. .
. m3 = 0, (). m3 , :
.
, , f (x).
_________________________________
.
W , . , , ᒺ .
, . , ,
n (X1, X2, , Xn) n- , n , n- .
_________________________________
31. .
Y = yi , X = xj .
:
X=xj Y=yi | x1 | x2 | xm | pyi | |
y1 | p11 | p12 | p1m | py1 | |
y2 | p21 | p22 | p2m | py2 | |
y3 | p31 | p32 | p3m | py3 | |
yk | pk1 | pk2 | pkm | pym | |
pxj | px1 | px2 | pxm |
:
_________________________________
32. .
ϳ . :
Κ = 0 Y .
ҳ :
|
|
, .
Y , Κ = 0 r = 0. г r , .
Y , r = 0, , r ¹ 0.
_________________________________
33. .
(, Y) , , (X < x) I (Y < y):
F(x,y) = P((X < x) I (Y < y)).
F(x, y)
1. 0 £ F(x, y) £ 1, 0 £ P((X < x) I (y < y)) £ 1.
2. F(x, y) + , , + , :
3.
4.
5. F(x, y) .
( < < b, Y < y) = F(b, y) F(a, y);
P(X < x, c < Y < d) = F(x, d) F(x, c).
6. (, Y) (a < X< b, c < Y < d) :
P(a < x < b, c < y < d) = F(b, d) + F(a, c) F(a, d) F(b, c).
_________________________________
34. ٳ .
.
f (x, y) , F (x, y) .
f (x, y)
1. f (x, y) ³ 0, F(x, y) .
2. (, Y) :
.
3. (, ) :
4.
5. ,
_________________________________
35. (,).
_________________________________
36. .
(, Y) (a < X< b, c < Y < d) :
P(a < x < b, c < y < d) = F(b, d) + F(a, c) F(a, d) F(b, c).
.
䳿:
A = (X < b, Y < d); B = (X < a, Y < c); C = (a < X < b, Y < c); D = (X < a, c < Y < d); E = (a < X < b, c < Y < d).
䳿 B, C, D, E , :
A = B U C U D U E.
P(A) = P(B U C U D U E) = P(B) + P(C) + P(D) + P(E).
P(x < b, y < d) = P(x < a, y < c) + P(a < x < b, y < c) + P( < a, c < < d) + P(a < x < b, c < y < d).
F(b, d) = F(a, c) + F(b, c) F(a, c) + F(a, d) F(a, c) + P(a<X<b,c<Y<d);
P(a<X<b,c<Y<d)=F(b,d)+F(a,c)F(a, d) F(b, c), .
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37. 1- .
Y, Y = = () , = , . , Y = () .
, Y = () .
1) :
= i | x1 | x2 | ............ | xk |
P(X = xi) = pi | p1 | p2 | ............. | pk |
Y = () :
|
|
Y = α (i) | α (1) | α (2) | .......... | α (k) |
P(Y = α (i) = i | p1 | p2 | ......... | pk |
f ():
.
f ()
.
_________________________________
38. 1- .
Y, Y = = () , = , . , Y = () .
, Y = () .
:
;
_________________________________
39. 2- .
Y
,
.
Y , Z . Y , Z .
_________________________________
40. .
.
( + Y) = () + (Y). (1)
1.
(+Y+)=()+(Y)+.
, , .
2.
.
_________________________________
41. .
ֳ X , :
, :
.
.
:
.
;
.
_________________________________
42. . г .
ֳ , :
. (1)
:
.
:
.
_________________________________
43. .
,
f () = ,
= (X), s = s (X). , s .
F(x)= dx. (2)
==.
: N (a; s).
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