Contents
Why is sigma algebra used in probability?
Sigma algebra is necessary in order for us to be able to consider subsets of the real numbers of actual events. In other words, the sets need to be well defined, under the conditions of countable unions and countable intersections, for it to have probabilities assigned to it.
What is a sigma field in probability?
A sigma-field refers to the collection of subsets of a sample space that we should use in order to establish a mathematically formal definition of probability. The sets in the sigma-field constitute the events from our sample space.
What is the difference between field and Sigma field?
The difference is in one condition. In Sigma-field you need being closed in respect of countable(finite and infinite countable) union but in field (without sigma) you only need being closed in respect of finite union. Here there is an example which is field but not sigma-field.
Why is sigma algebra important in probability theory?
Sigma algebra is considered part of the axiomatic foundations of probability theory. The topic is briefly covered in Casella & Berger’s Statistical Inference. The need for sigma algebras arises out of the technical difficulties associated with defining probabilities.
How are σ algebras used to calculate probabilities?
If we are able to think of ‘measure’ as a ‘probability’ for a moment (this will be formalised below) and sets in the σ − algebra as ‘events’ that we might be interested in calculating probabilities for then we can see that the definition of a σ − algebra lets us assign unambiguous probabilities to those events.
What is the definition of a probability space?
To define something in Probability as measurable we need to be able to mathematically define a Probability Space. A Probability Space is also referred to as a Probability Triple and consists, unsurprisingly, of 3 parts: \\Omega Ω – This is just the set of outcomes that we are sampling from.
Which is the Triple of the sigma algebra?
This finally allows us to define a probability space as the triple ( Ω, F, P). The sets within the σ − algebra F are known as events. In future articles we will utilise probability spaces to define another important concept in probability theory, namely the Random Variable.