Random structure function

The random structure function is the third component of the Bernoulli space which constitutes the stochastic model within Bernoulli stochastics. 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 $X|\{d\}$ 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 $\mathfrak{𝔓}(\{d\})$ of the singletons $\{d\}$ as follows:

$\mathfrak{𝔓}({\mathfrak{𝔇}}_0)=\frac{1}{|{\mathfrak{𝔇}}_0|}\sum _{d\in {\mathfrak{𝔇}}_0}\mathfrak{𝔓}(\{d\})$

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

$P_{X|{\mathfrak{𝔇}}_0}(E)=\frac{1}{|{\mathfrak{𝔇}}_0|}\sum _{d\in {\mathfrak{𝔇}}_0}{P}_{X|\{d\}}(E)$

Thus, the probability distribution of the random variable $X|{\mathfrak{𝔇}}_0$ is given by the mean of the probability distributions of the random variables $X|\{d\}$.

References

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