| Meaning: |
The nested distribution gives the distribution of a sum of random
variables \(T_1, T_2, ..., T_N\) that each are described by a continuous
("inner") distribution. The number of addends \(N\) itself is a random
variable with discrete ("outer") distribution. In literature, this
distribution is also denoted as compound distribution. |
| Parameters: |
-
inner distribution with PDF \(g(t)\) and DF \(G(t)\)
-
outer distribution \(p_i\)
|
| PDF: |
\(P(T=t) = f(t) = \sum\limits_{n=0}^{\infty} p_n \cdot (g_1(t) \otimes
... \otimes g_n(t)) \mbox{ with } g_i(t) \equiv g(t) \forall i\) |
| Expected value: |
\(E[T] = E[N] \cdot E[T_i]\) |
| Variance: |
\(VAR[T]= VAR[T_i] \cdot E[N] + VAR[N] \cdot (E[T_i])^2\) |
| Coefficient of variation: |
\(c_T = \sqrt{\frac{c_{T_i}^2}{E[N]} + c_N^2}\) |
| LST: |
\(\phi(s) = H(\Psi(s))\) , if \(H(z)\) represents the generating function
of the external and \(\Psi(s)\) the LST of the inner distribution. |
| Parser example: |
[...].Distribution = NestedDistribution
[...].Distribution.InnerDist = Constant
[...].Distribution.InnerDist.Mean = 2.3
[...].Distribution.OuterDist = Geometric
[...].Distribution.OuterDist.Mean = 9
|