Difference between revisions of "Probability space"

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==Definition==
 
==Definition==
Given a [[Measure space|measure space]] {{M|(X,\mathcal{A},\mu)}}  
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Given a [[Measure space|measure space]] {{M|(X,\mathcal{A},\mu)}} where <math>\mu</math> is a [[Probability measure]]<ref>p22 - Measures, Integrals and Martingales - Rene L. Schilling</ref>, we recall that <math>\mu(X)=1</math> (and we will write {{M|\mu}} as {{M|\mathbb{P} }} in line with notation).
  
We call it a probability space if <math>\mu</math> is a '''Probability measure'''<ref>p22 - Measures, Integrals and Martingales - Rene L. Schilling</ref>, which means that <math>\mu(X)=1</math>
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Now {{M|(X,\mathcal{A},\mathbb{P})}} is a ''Probability space''
  
  

Latest revision as of 22:55, 2 May 2015

Definition

Given a measure space [ilmath](X,\mathcal{A},\mu)[/ilmath] where [math]\mu[/math] is a Probability measure[1], we recall that [math]\mu(X)=1[/math] (and we will write [ilmath]\mu[/ilmath] as [ilmath]\mathbb{P} [/ilmath] in line with notation).


Now [ilmath](X,\mathcal{A},\mathbb{P})[/ilmath] is a Probability space


A Probability space is usually denoted [ilmath](\Omega,\mathcal{A},\mathbb{P})[/ilmath], here:

Name Symbol Type Description
  • State space
  • Sample space
[ilmath]\Omega[/ilmath] Set All the different states one can have or samples one can take
  • Event space
[ilmath]\mathcal{A} [/ilmath] [ilmath]\sigma[/ilmath]-algebra The events we can have
  • Probability measure
[ilmath]\mathbb{P} [/ilmath] Function [math]\mathbb{P}:\mathcal{A}\rightarrow[0,1]\subset\mathbb{R}[/math] Assigns probabilities to events

Example

Discrete probability space

Let us consider two die being thrown as our state or sample space - I prefer state because it is the set of states the experiment may take.

Then:

Part of prob. space Definition Comment
State space [math]\begin{array}{lr} \Omega=\{ & (1,1), & (1,2), & \cdots &, (1,6),\\ & (2,1), & (2,2), & \cdots &, (2,6), \\ & \vdots \\ & (6,1), & (6,2), & \cdots &, (6,6) & \} \end{array}[/math] The set of all possible states, there are 36 all-together.
Event space [math]\mathcal{P}(\Omega)[/math] (see power set)

= all subsets of [ilmath]\Omega[/ilmath]

Union works as or, for example [math]\{(1,2),(3,4)\}[/math] is the event that we get [ilmath](1,2)[/ilmath] or [ilmath](3,4)[/ilmath]
Probability measure [math]\mathbb{P}:\mathcal{P}(\Omega)\rightarrow[0,1]\subset\mathbb{R}[/math] where:

[math]\mathbb{P}(A)\mapsto \frac{1}{36}|A|[/math]

Clearly [math]\mathbb{P}(\Omega)=1[/math] and this is a measure!

This example alone isn't very interesting, it becomes interesting when one considers the random variable which could be for example the sum of the values shown on the die. That example is used on the random variable page.

Continuous probability space


TODO: Think of example - Normal?



See also

References

  1. p22 - Measures, Integrals and Martingales - Rene L. Schilling