, λ(t). , (t, t + ∆ t), , :
λ(t) = = m ′(t),
m (t) (0, t). λ(t) .
, , , λ(t), τ > 0, t 0 ( , , t 0 + τ) (t 0, τ) , t 0 t 0 + τ.
1. λ(t):
1) (t 0, τ)
m (t 0, τ) = , (m = 1, 2, ),
m (t 0, τ) , [ t 0, t 0 + τ ] m , t 0 t 0 + τ,
= ;
2) (t 0, τ)
D ( (t 0, τ)) = a.
, , , , .
. p 1(t 0; Δ t) ( ) t 0 t 0 + Δ t.
, Δ t, , t 0.
2. t 0 t 0 + Δ t λ(t)
p 1(t 0; Δ t) ≈ λ(t 0) ∙ Δ t (Δ t → 0).
, .
(t 0, τ).
|
λ (t) | ||
(t 0, τ) | =a (t 0, τ) = M( (t 0, τ)) = | |
(t 0, τ) | m (t 0, τ) = , (m = 1, 2, ) | |
, t 0 t 0 + τ | p0 (t 0, t 0 + τ) = ( (t 0, τ)= 0) = | |
, t 0 t 0 + τ k (k =1,2,3) | ( (t 0, τ) <k) = | |
, t 0 t 0 + τ k (k =1,2,3) | ( (t 0, τ) ≥ k) = 1- | |
, t 0 t 0 + τ | ( (t 0, τ) ≥ 1) = 1 - | |
t 0 t 0+ Δ t | p 1(t 0; Δ t) ≈ λ(t 0) ∙ Δ t (Δ t → 0). | |
(t 0, τ) | D ( (t 0, τ)) = | |
(t 0, τ) | σ( (t 0, τ)) = = |
|
|
. , , 0 t . , λ(t). , t 0, (t 0)
(t) = P (T < t) = 1 P (T ≥ t).
P (T ≥ t) , t 0 t 0 + t
P ( (t 0) ≥ t) = ℮- = , (1)
(t) = 1 - .
,
(t) = λ(t 0 + t) (t > 0).
. t 0 λ(t).
, , , : .
.
, , .
. ( ), 1, 2, .
:
1) , . ; . ( ) . , .
2) ( ) V. , , , .. L . , , . , 1 = , 2 = , . , , , .
|
|
. .
(). , , , ( ). , .
, , , , - .
, .
( ).
, (1). , : , 1, 2, , .
, (k +1)- , k- ( k).
k - . 0 t 1, 2, . k +1
= ,
1, 2, k ,
f (t) = λ ℮- λt (t > 0). (1)
, (t, t + dt), k -1 (0, t). λdt;
pk (t) = .
,
fk (t) dt = λ dt,
fk (t) = (t > 0).
k- . , k = 1 (1).
fk (t) mk Dk.
mk = = km 0,
m 0 = .
mk = .
Dk = , σk = .
λk k - ( k ). λk = , ,
λk = λ = kλk.
, λk k - k λ .
k - k. λk k , λ , k 0 t, .. k k. k- .
:
1) = kλk = λ, k = 1,2.3,
2) = , k = 1,2.3,.
|
|
3)
() = mk = , k = 1,2.3,.
4)
D () = Dk = = , k = 1,2.3,.
5)
σ () = , k = 1,2.3,.
, : k , .
; , k, : (k = 1) (k = ∞). , , .
. :
mt = 2 (), Dt = 0,8 (2).
.
. λ = = 0,5.
k ≈ = = 5.
, .
k - k = k - .
|
k - k | - k - | ||
λ | λ | ||
k - | λk | = kλk = λ | |
fk (t) = | = k fk (kt) = | ||
λ = kλk | kλ | ||
() = = | () = = | ||
D () = = | D () = | ||
σ (T) = = | σ () = |
1. , ,
1) .
2) .
35 . .
. 1) = ≈ 0,6886.
2) = ≈ 0,1738.►
2. , , . 4 5 , 1 . 5 , 2 . 6 .
1) , .
2) .
. () = 6 , , λ = .
1) 4 , , .
|
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= ≈ 0,4883.
2) , 4 5 , , ..
F (5) F (4) = - = - ≈ 0,5117 0,4361 = 0,075.
, 5 ,
≈ 0,4361.
,
10,075 + 20,4361 = 0,9472 . ►
3. , , .
, τ , τ, ; ; . , , 6.
, :
1) 7 ;
2) 7 ;
3) 7 ;
4) ;
5) ;
6) ;
6) .
. 1
p 7(1) ≈ 0,138;
( (1) < 7) ≈ 0,606;
( (1) ≥ 7) ≈ 0,394;
p 0() ≈ 0,223;
( () ≥ 1) ≈ 0,950;
P (T < ) ≈ 0,670;
P (T ≥ ) ≈ 0,330.►
4. , , . , , τ, , . , , , - .
.
, , : λ(t) = .
1) 6 ;
2) 6 ;
3) 5 ;
4) ;
5) ;
6) , ;
7) , .
. .
6(0; 4) ≈ 0,072;
6(4; 4) ≈ 0,005;
( (8,4) ≥ 5) ≈ 0,999;
p0 (0,2) ≈ 0,022;
( (5,2) ≥ 1) ≈ 0,999;
P (T (9) ≥ ) ≈ 0,147;
P (T (6) < ) ≈ 0,665.►
5. . . : 2,5 (), 2,7 (2).
, , , k . . , .
. k = 2; = 0,4 ( )
|
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= 0,64∙ t ∙℮-0,8 t; ≈ 0,5.►