Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
.pdf
N |
n1 |
n2 |
x1 |
x2 |
σ12 |
σ22 |
α |
|
|
n1 n2 |
x¯1, x¯2 |
σ12, σ22 |
|
α |
|
n = 120 |
σ = 5
x¯ = 23, 54
H0 : a = a0 = 23 
H1 : a = 23
H0 : p = p0 = 0, 12
H1 : p > 0, 12
α = 0, 05
α = 0, 05
ξ
\
\
\
\
\
\
ξ(t) |
|
t |
t |
ξ(t) = ζ ·sin t 
t 0 ζ N (2; 1) 
t
t
|
ξ(t) |
|
x(t) |
ξ(1) = ζ · sin 1 |
t = 1 |
ζ |
|
|
−3 |
x1(t) = 2 sin t |
x2(t) = 3 sin t |
m(t) |
t |
mξ (t) = M ξ(t) . |
|
m(t) 
ξ(t) 
f (t) 
m(t) 
M f (t) |
|
= f (t) |
|
|
|
|
|
|||
M |
|
f (t) |
|
ξ(t) = f (t) M ξ(t) |
|
|||||
|
|
± |
|
|
ξ1(t) |
|
± |
|||
M ξ1(t)· |
|
ξ2(t) = M· |
|
M ξ2(t) |
||||||
σ2(t) 
t 
σξ2(t) = D ξ(t) .
"
σξ (t) = D ξ(t)
σξ2(t) 
σξ2(t) 0
D f (t) = 0
D f (t) · ξ(t) = f 2(t) · D ξ(t)
D ξ(t) ± f (t) = D ξ(t)
Kξ(t1; t2)
ξ(t1) 
ξ(t2) 
Kξ(t1; t2) = M ξ(t1) − m(t1) · ξ(t2) − m(t2) .
◦
ξ(t) = ξ(t) − m(t),
◦ ◦
Kξ(t1; t2) = M ξ(t1) · ξ(t2) .
σ2 |
|
|
|
|
|
|
|
|
mξ (t) |
(t) Kξ(t1; t2) |
|
|
|
|
|
|
|
|
|
ξ |
|
|
|
|
|
|
|
|
|
|
ζ N (2; 1) |
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
mξ (t) = M ξ(t) |
= M (ζ · sin t) = M (ζ) · sin t = 2 sin t, |
|||||||
|
σξ2(t) = D ξ(t) = D(ζ |
sin t) = D(ζ) sin2 t = sin2 t, |
|
||||||
|
Kξ(t1; t2) = M (ζ |
sin t1 |
·2 sin t2) (ζ sin· t2 |
− |
2 |
|
sin t2) |
= |
|
|
= M (ζ |
2)2 sin |
·t1 sin t2− = D(ζ) sin· t1·sin t2 = sin t1 sin |
t2. |
|||||
|
mξ (t) =−2 |
sin t |
σ2(t) = sin2 t Kξ(t1; t2) = sin t1 |
· |
sin t2 |
|
|||
|
|
|
ξ |
|
|
|
|
|
|
Kξ(t1; t2) = Kξ(t2; t1) 
Kξ(t; t) = σξ2(t) 
|Kξ(t1; t2)| σξ (t1) · σξ (t2) 
Kξ(t1; t2)
ρξ (t1; t2) = σξ (t1) · σξ (t2) .
ρξ (t1; t2) = ρξ (t2; t1) 
ρξ (t; t) = 1 
|ρξ (t1; t2)| 1
ξ(t) 
ζ(t) 
Rξζ (t1; t2)
ξ(t1) 
ζ(t2) 
|
|
|
◦ |
|
◦ |
|
|
Rξζ (t1; t2) = M ξ(t1) |
|
ζ(t2) . |
|||
Rξζ (t1; t2) ≡ 0 |
ξ(t) ζ(t) |
|
|
· |
|
|
t1, t2 |
|
|
|
|
|
|
Rξζ (t1; t2) 
Rξζ (t1; t2) = Rζξ (t2; t1) 
|Rξζ (t1; t2)| σξ (t1) · σζ (t2) 
|
Rξξ(t1; t2) = Kξ(t1; t2) |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
Kξ(t1; t2) |
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
ξ(t) = ξ1(t) + ξ2(t) |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
Kξ(t1; t2) = Kξ1 (t1; t2) + Kξ2 (t1; t2) + Rξ1ξ2 (t1; t2) + Rξ2ξ1 (t1; t2). |
|
|
|||||||||||||
|
|
◦ |
◦ |
|
|
|
|
◦ |
◦ |
|
|
◦ |
◦ |
|
|
|
Kξ(t1; t2) = M ξ(t1) |
ξ(t2) = M ξ (t1) + ξ (t1) |
◦ · |
|
ξ (t2) + ξ2 |
(t2) = |
|||||||||||
◦ |
◦ |
|
· ◦ |
|
◦ |
|
|
1 |
◦ 2 |
|
1 |
◦ |
|
|
◦ |
|
= M ξ1(t1) · ξ1(t2) + M ξ2(t1) · ξ2(t2) + M ξ1(t1) · ξ2(t2) + M ξ2(t1) · ξ1(t2) =
|
+ R (t ; t ) + R |
(t ; t ). |
|
= Kξ1 (t1; t2) + Kξ2 (t1; t2) ξ1ξ2 1 2 |
ξ2ξ1 1 2 |
Kξ(t1; t2) = Kξ1 (t1; t2) + Kξ2 (t1; t2).
ξ1(t) 
ξ2(t) Rξ1ξ2 (t1; t2) ≡
≡ 0 
Rξ2ξ1 (t1; t2) = Rξ1ξ2 (t2; t1) ≡ 0 
t1; t2
ζ = ξ1 + ξ2i
ξ1 
ξ2 
i
|
ζ(t) |
|
ζ(t) = ξ1(t) + ξ2(t)i, |
ξ1(t) ξ2(t) |
i |
(i2 = −1) |
|
ζ(t) = ξ1 + ξ2(t)i
ξ2(t) |
0 |
|
mξ |
(t) = M ξ1(t) + M ξ2(t) i. |
|
≡
ξ1(t) 
◦
ζ(t) = ξ(t) − mξ (t) 
|
◦ |
. |
σζ2 |
(t) = M |ζ (t)|2 |
|
ζ2(t) ≡ 0 |
|
|
|
|
σ2 |
(t) = σ2 |
(t) + σ2 |
(t) |
|
|
|
|
|
|
|
|
|
||||
|
|
ξ |
|
|
ξ1 |
|
|
ξ2 |
|
|
|
|
|
|
|
|
|
|
|
◦ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
◦2 |
|
◦2 |
(t) |
|
|
|
|
|
|
|
|
|
|
||
|
|ζ (t)| = "ξ1 |
(t) + ξ2 |
|
|
|
|
|
|
|
|
|
|
||||||
2 |
|
◦ |
|
|
◦2 |
◦ |
2 |
(t) |
◦2 |
(t) |
◦2 |
(t) |
2 |
2 |
|
|||
σζ |
= M |
|ζ (t)|2 |
= M ξ1 |
(t) + ξ2 |
= M ξ1 |
+ M ξ2 |
= σξ1 |
(t) + σξ2 |
(t). |
|||||||||
◦ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
◦ |
|
|
|
|
|
|
|
|
||
ζ (t1) = ξ1(t1) + ξ2(t1)i ζ (t2) = ξ1(t2) − ξ2(t2)i |
|
|
|
|
||||||||||||
|
|
|
|
|
|
|
|
|
|
◦ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
Kζ (t1; t2) = M ζ (t1) · ζ◦ (t2) . |
|
|
|
◦ |
|||||
|
|
|
|
|
◦ |
|
|
|
|
|
|
|
Kζ (t; t) = M ζ(t) · |
|||
· ζ◦ |
(t) |
|
= σζ2(t). |
|
|
|
||||||||||
= M |ζ(t)|2 |
|
|
|
|
ζ(t) |
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Kζ (t1 |
; t2) = Kξ (t1; t2) + Kξ2 (t1 |
; t2) + Rξ2ξ1 (t1; t2) |
|
Rξ1ξ2 (t1 |
; t2) i. |
||||||||
|
|
|
|
|
|
1 |
ξ1(t) ξ2(t) |
|
|
|
− |
|
|
|
||
Kζ (t1; t2) = Kξ1 (t1; t2) + Kξ2 (t1; t2).
ξ(t)
mξ (t) 
t
Kξ(t1; t2) 
mξ (t) = m, Kξ (t1; t2) = kξ (t2 − t1).
Kξ (t1; t2) = kξ (t2 − t1) = kξ (τ ), |
τ = t2 − t1. |
kξ (−τ ) = kξ (τ ).
Kξ (t1; t2) 
kξ (t1; t2) = kξ (t2; t1) kξ (−τ ) = kξ (t1 − t2) = Kξ (t2; t1) =
= Kξ (t1; t2) = kξ (t2 − t1) = kξ (τ ).
σξ2(t) = kξ (0) = σξ2.
Kξ (t1; t2) 
σξ2(t) = Kξ (t; t) = kξ (t − t) = kξ (0) = 
.
|kξ (τ )| kξ (0).
Kξ (t1; t2) 
" "
|Kξ (t1; t2)| Kξ (t1; t1) · Kξ (t2; t2) |kξ (τ )| kξ (0) · kξ (0)
|kξ (τ )| kξ (0) |kξ (τ )| σξ2.
ρ
j, k 
M 
j 
M