Random structure function

The random structure function^[1] is the third component of the Bernoulli space which constitutes the stochastic model within Bernoulli stochastics.^[2] The Bernoulli space describes the transition from past to future. The determinate past is represented by a variable D which is called deterministic variable, because its value is fixed. The future represented by the variable X is subject to randomness and X is therefore called random variable. The random variable X may adopt one of a set of different values according to a random law which depends on the actual initial conditions given by the value d of the deterministic variable. The random law does not only fix the range of variability of X but also the probability of the future events which are given by subsets of the range of variability of X.

Probability distribution

The random variable X stands for the future indeterminate outcome of a process. If the process is repeated then different outcomes will occur according to a random law that depends on the actual initial conditions given by the value d of the deterministic variable D. The random variable X under the condition d is denoted ${\displaystyle X|\{d\}}$ where the set of possible initial conditions is given by the ignorance space ${\mathfrak {D}}$. The random structure function assigns to each subset of the ignorance space a probability distribution.

Let $\mathfrak{D}_0$ be a subset of the ignorance space ${\mathfrak {D}}$ then the corresponding probability distribution is obtained from the images ${\displaystyle {\mathfrak {P}}(\{d\})}$ of the singletons ${\displaystyle \{d\}}$ as follows:

${\displaystyle {\mathfrak {P}}({\mathfrak {D}}_{0}))={\frac {1}{|{\mathfrak {D}}_{0}|}}\sum _{d\in {\mathfrak {D}}_{0}}{\mathfrak {P}}(\{d\})}$

It follows that for any future event E, we have:

${\displaystyle P_{X|{\mathfrak {D}}_{0}}(E)={\frac {1}{|{\mathfrak {D}}_{0}|}}\sum _{d\in {\mathfrak {D}}_{0}}P_{X|\{d\}}(E)}$

Thus, the probability distribution of the random variable ${\displaystyle X|{\mathfrak {D}}_{0}}$ is given by the mean of the probability distributions of the random variables ${\displaystyle X|\{d_{0}\}}$.

References

1. Elart von Collani, Defining and Modeling Uncertainty, Journal of Uncertainty Systems, Vol. 2, 202−211, 2008, .
2. Elart von Collani (ed.), Defining the Science Stochastics, Heldermann Verlag, Lemgo, 2004.

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