Дронов С.В. Конспект лекций по теории случайных процессов
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X(t) = t + V cos t + U sin t, Y (t) = 2t
2
−V sin t + U cos t,
DU = DV = σ
2
Z(t) = X(t) + Y (t)
K
X,Y
(t, s) = cov(X(t), Y (s))
X(t) = x
1
t + x
2
sin t
(x
1
, x
2
)
(1, −1)
!
.
MX(t), K(t, s)
ϕ
K(t, s) = a
2
exp
n
−b
2
(t − s)
2
o
.
ξ(t)
Mξ(t) = 0
t
X(s) = ξ(t)ξ(t + s).
0 0 1 0
1 0 0 0
1/2 1/2 0 0
1/3 1/3 1/3 0
.
0 1 0 0
0 0 0 1
0 1 0 0
1/3 0 2/3 0
.
q p = 1 − q p
(n)
00
n
N
m
P =
q p 0 0 ... 0
q 0 p 0 ... 0
0 q 0 p ... 0
... ...
0 0 0 0 q p
.
lim
n→∞
P
n
Γ
i, k ∈ Γ
lim
n→∞
p
ik
(n) = 1, lim
n→∞
p
ki
(n) = 1.
X(t), t > 0
ξ
n
= X(n), n ∈ N
X(t)
MX(t) = 2t + 1; K(t, s) = exp{−(t − s)
2
}.
X(t) = e
−at
sin(ωt + ϕ),
a, ω ϕ
[0, 2π]
t > 0
X(t)
X(t) = 1 + t + Ut + V t
2
+ W t
3
,
DU = 2, DV = 1, DW = 0, 1
dX
dt
X(t) = x
1
t + x
2
sin t
(x
1
, x
2
)
2 1
1 3
.
Y (t) =
Z
t
0
X(s) ds.
X(t) 0, 2 cos
2
ωt
K(t, s) = 0, 4 cos ωt cos ωs
X(t) K(t, s)
Y (t) = X(t) +
dX
dt
.
Y
X(t) Y (t) =
R
t
0
X(s)ds
X(t) Y (t)
G t = 0 k
t
µt + o(∆t)
t
G
t
λt + o(∆t)
ξ(t), t ∈ [a, b]
X(t) =
Z
b
a
B(t − s)ξ(s) ds,
B
K(t) = e
−|t|
cos βt
f(λ) =
α
π(α
2
+ λ
2
)
.
ξ
t ξ
ξ
1
, ξ
2
, ... Mξ
i
=
1, i = 1, 2, . . .
X
n
=
n
Y
k=1
ξ
k
, n ∈ N −
ξ(t) Mξ
2
(t) < ∞
X(t) λ
Y (t) = exp{X(t) −at}
a < λt a > λt
ξ(t) a = Mξ(t)
