SKEDSOFT
Introduction:
Probabilistic model is the mathematical structure of any phenomenon whose input or output both are uncertain in nature they depend over some particular criteria or region.
Fundamentals of probabilistic models:
The following are two fundamental probabilistic models that can serve as building blocks for more complex models:
a) The uniform distribution on [0, 1], which assigns probability b−a to every interval [a, b] ⊂[0, 1].
b) A model of an infinite sequence of fair coin tosses that assigns equal probability, ½ n,to every possible sequence of length n.
1. These two models are often encountered in elementary probability and used without further discussion. Strictly speaking, however, we need to make sure that these two models are well-posed, that is, consistent with the axioms of probability.
2. To this effect, we need to define appropriate σ-fields and probability measures on the corresponding sample spaces. In what follows, we describe the required construction, while omitting the proofs of the more technical steps.
Caratheodor ´ y’s extension theorem:
1. The general outline of the construction we will use is as follows. We are interested in defining a probability measure with certain properties on a given measurable space (Ω, F).
2. We consider a smaller collection, F0 ⊂F, of subsets of Ω, which is a field, and on which the desired probabilities are easy to define.
3. Furthermore, we make sure that F0 is rich enough, so that the σ-field it generates is the same as the desired σ-field F. We then extend the definition of the probability measure from F0 to the entire σ-field F.
4. This is possible, un der few conditions, by virtue of the following fundamental result from measure theory.
Lévesque measure on [0, 1] and on r:
we construct the uniform probability measure on [0, 1], also known as Lévesque measure. Under the Lévesque measure, the measure as signed to any subset of [0, 1] is meant to be equal to its length. While the definition of length is immediate for simple sets (e.g., the set [a, b] has length b − a), more general sets present more of a challenge.
