..
:
:
() 2
.
1. .3-13
2. 14
, , .
( ). X, Y, Z, ..
, , , .
. . , Z: 1, 2, 3, 4, 5, 6.
. n . : 0, 1, 2, , n .
.
, .
, , .
, , .
, . .
.
, ( ) :
1 | 2 | n | ∑ | ||
1 | 2 | n |
. , : 1, 2, , n . . = i (i = 1, 2, , n) . , 1 + 2 + + n = 1.
, , . . () .
F(x), , , , ..
F(x) = P (X < x).
F(x) .
:
1. [0,1]: 0 ≤ F(x) ≤ 1
2. .
3. , , (, b), : ( < X < b) = F(b) F().
4. (, b), F(x) = 0 ≤ ; F(x) = 1 ≥ b.
5. :
, 1, 2, , n,
F(x)=
, i, .
F(x). , - , , xi < x ≤ xi +1. , 1, 2, 3, , i. < x , , k = 1, 2, , i. , < x , , , = 1, = 2, = 3, , = i. ,
P(X< x) = P(X= )+P(X= )++ P(X= ) =
, F(x)
F'(x)= φ(x).
φ(x) ( ) .
φ(x) F(x), : φ(x)≥0. , .
F(x) φ(x), - .
P(a ≤ X ≤ b) =
= ∞ b = +∞, (∞, +∞), . ,
, (, b), . = ∞, b = t, :
F(x) = P ( ∞ < X < x) = .
|
|
. ; . , .
1. ( o, e, ()).
2. ( D(X), σ()).
3. y = φ(x) ( As, ).
.
, . , , :
M(X)=
, φ(x) :
M(X)=
, :
|
|
, .. .
:
1. () = C, = const;
2. M(C ∙ ) = ∙ ();
3. ( Y) = () (Y), X Y ;
4. ( ∙ Y)= ()∙ (Y), X Y .
, , .
, , (. 2.3), , (. 2.4).
. 2.3 . 2.4
, , , ..
( < ) = (X > )
, ( < ) = 0,5, .. F () = 0,5. , φ(x) , (. 2.5). (. 2.6).
. 2.5 . 2.6
.
D(X) = M(X ())2.
:
)
D(X)=M(;
)
D(X) = f()d x [M(X)]2 .
:
1. D(C) = 0, = const;
2. D(C × X) = C2∙ D(X);
3. D (X Y) = D (X) + D (Y), X Y .
, ..
σ(X) = .
.
k- α k k, .. α k = (k).
.
k- μ k ( ()) k , .. μ k = ( ()) k.
.
α k = , μ k = , i = p(X = xi). :
α k = , μ k = ,
φ(x) .
As = μ3 / σ3 .
, m3 . (.2.7) (). As , , , (.2.7) . ( ).
. 2.7
k
k = μ4 / σ4 3.
, , , μ4 / σ4 = 3. , . , , , k > 0, k < 0 (.3.8).
. 2.8
|
|
n (, q =1 p). , .
, n , 1 , 2 , , n . , : 1 = 0, 2 = 1, 3 = 2,, n +1 = n. = k ( k ) :
Pn(k) = C nk pk qn k ,
k = 0, 1, 2, , n.
, , :
K | n | |||||
Cn 0 p 0 qn | Cn 1 p 1 qn 1 | Cnk pk qnk | Cnn pn q 0 |
, Pn (k) k = 0, 1, 2, , n
(p + q)n = qn + Cn 1 p 1 qn 1 + + Cnk pk qn k + + pn.
, 1, p + q =1.
[ a, b ], :
. 2.9.
φ()
. 2.9
. , , 1,
= 1/(b a).
[ α, β ] Ì [ a, b ]. , p=P( ..
,
L ( ) [ a, b ] l [ α, β].
, .. [ a, b ], . , [ α, β] Ì [ a, b ]. [ α, β] .
: , , L , :
.
().
, .
,
, , , . λ .
. (, , , ..). , . . 2.11.
. 2.11
|
|
( ), :
,
σ >0.
( ). (. 2.12) = , , = σ .
. 2.12
, ( ), σ . : As = Ex = 0.
, σ . . , ( ) , (. 2.13).
σ . , , . , , 1, σ , σ = (. 2.14).
. 2.13 . 2.14
φ() = 0, σ = 1 , .
, . 2.15.
, U= , D(U)=1, M (U) = 0. U= ,
, σ.
F(x) =
, , [ α, β]
, .
, . , ( ). , .. .
. , , , .