Variability function

The variability function is the second 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 and, therefore, determinate. 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 D.

Range of future variability

The random variable X stands for the indeterminate future 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 set of values which may be adopted by the random variable $X|\{d\}$ is given by the variability function denoted ${\mathfrak {X}}$, which is defined on the subsets of the ignorance space ${\mathfrak {D}}$ and assigns to each subset of the ignorance space the corresponding range of variability of $X|\{d\}$.

Let ${\mathfrak {D}}_{0}\epsilon {\mathfrak {D}}$ then the corresponding range of variability is obtained from the images of the singletons as follows:

${\mathfrak {X}}({\mathfrak {D}}_{0})=\cup _{d\epsilon {\mathfrak {D}}_{0}}{\mathfrak {X}}(\{d\})$

The range of variability of a random variable contains all those values which may be adopted by the future outcome. It follows that the range of variability is of utmost importance for any decision making process. If no risk is admitted, then each of the values of the range of variability must be equally considered.

References

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