Cлучайная величина
ξ
{\displaystyle \xi}
гиперэкспоненциальное распределение с параметрами
(
m
;
α
1
,
α
2
,
…
,
α
m
;
λ
1
,
λ
2
,
…
,
λ
m
)
,
α
k
λ
k
≥
0
,
∑
k
=
1
m
α
k
=
1
{\displaystyle (m; \alpha _1, \alpha _2, \ldots, \alpha _m; \lambda _1, \lambda _2, \ldots, \lambda _m), \alpha _k \lambda _k \geq 0, \sum _{k=1}^m \alpha _k =1}
f
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x
)
=
{
∑
k
=
1
m
α
k
λ
k
e
−
λ
k
x
,
x
≥
0
0
,
x
<
0
.
{\displaystyle
f(x) = \left\{ \begin{matrix} \sum_{k=1}^m \alpha _k \lambda _k e^{-\lambda _k x}, & x \geq 0 \\ 0, & x < 0 \end{matrix} \right..
}
Характеристическая функция [ ]
φ
(
t
)
=
∑
k
=
1
m
α
k
λ
k
λ
k
−
i
t
.
{\displaystyle
\varphi (t) = \sum_{k=1}^m \frac{\alpha _k \lambda _k}{\lambda _k - it}.
}
Свойства [ ]
E
ξ
k
=
∑
l
=
1
m
α
l
k
!
λ
l
k
;
{\displaystyle \mathbf{E}\xi ^k = \sum_{l=1}^m \alpha _l \frac{k!}{\lambda _l^k} ;}
D
ξ
=
2
∑
l
=
1
m
α
l
λ
l
2
−
(
∑
l
=
1
m
α
l
λ
l
)
2
.
{\displaystyle
\mathbf{D}\xi=2\sum_{l=1}^m \frac{\alpha _l}{\lambda _l^2} - \left( \sum_{l=1}^m \frac{\alpha _l}{\lambda _l} \right) ^2.
}
Значение [ ] Введем случайные величины
I
k
(
k
=
1
,
…
,
m
)
{\displaystyle I_k (k = 1, \ldots, m)}
P
[
I
k
=
1
]
=
α
k
{\displaystyle \mathbf{P} [I_k = 1] = \alpha _k}
P
[
∑
k
=
1
m
I
k
=
1
]
=
1
{\displaystyle \mathbf{P} \left[ \sum_{k=1}^m I_k = 1 \right] = 1}
ξ
k
(
k
=
1
,
…
,
m
)
{\displaystyle \xi _k (k = 1, \ldots, m)}
λ
k
{\displaystyle \lambda_k}
I
k
{\displaystyle I_k}
ξ
=
∑
k
=
1
m
I
k
ξ
k
{\displaystyle \xi = \sum_{k=1}^m I_k \xi _k}
(
m
;
α
1
,
α
2
,
…
,
α
m
;
λ
1
,
λ
2
,
…
,
λ
m
)
{\displaystyle (m; \alpha _1, \alpha _2, \ldots, \alpha _m; \lambda _1, \lambda _2, \ldots, \lambda _m)}
См.также [ ] 
![{\displaystyle \mathbf {P} [I_{k}=1]=\alpha _{k}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/c748906c1de98d9fe68ea1304c334cb58d7aa436)
![{\displaystyle \mathbf {P} \left[\sum _{k=1}^{m}I_{k}=1\right]=1}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/cc16e800f7efb4825e0cd2d20d4df25c0367e0b9)
