Difference between revisions of "Probability space"
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==Definition== | ==Definition== | ||
| − | Given a [[Measure space|measure space]] {{M|(X,\mathcal{A},\mu)}} | + | 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). |
| − | + | Now {{M|(X,\mathcal{A},\mathbb{P})}} is a ''Probability space'' | |
| + | |||
| + | |||
| + | A '''Probability space''' is usually denoted {{M|(\Omega,\mathcal{A},\mathbb{P})}}, here: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Name | ||
| + | ! Symbol | ||
| + | ! Type | ||
| + | ! Description | ||
| + | |- | ||
| + | | | ||
| + | * State space | ||
| + | * Sample space | ||
| + | | {{M|\Omega}} | ||
| + | | Set | ||
| + | | All the different states one can have or samples one can take | ||
| + | |- | ||
| + | | | ||
| + | * Event space | ||
| + | | {{M|\mathcal{A} }} | ||
| + | | [[Sigma-algebra|{{sigma|algebra}}]] | ||
| + | | The events we can have | ||
| + | |- | ||
| + | | | ||
| + | * Probability measure | ||
| + | | {{M|\mathbb{P} }} | ||
| + | | 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|die]] being thrown as our ''state'' or ''sample'' space - I prefer ''state'' because it is the set of states the experiment may take. | ||
| + | |||
| + | Then: | ||
| + | |||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! 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|power set]])<br/> | ||
| + | = all subsets of {{M|\Omega}} | ||
| + | | Union works as or, for example <math>\{(1,2),(3,4)\}</math> is the event that we get {{M|(1,2)}} or {{M|(3,4)}} | ||
| + | |- | ||
| + | | Probability measure | ||
| + | | <math>\mathbb{P}:\mathcal{P}(\Omega)\rightarrow[0,1]\subset\mathbb{R}</math> where:<br/> | ||
| + | <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|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== | ==See also== | ||
| Line 10: | Line 75: | ||
==References== | ==References== | ||
| + | <references /> | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
Latest revision as of 22:55, 2 May 2015
Contents
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 |
|---|---|---|---|
|
[ilmath]\Omega[/ilmath] | Set | All the different states one can have or samples one can take |
|
[ilmath]\mathcal{A} [/ilmath] | [ilmath]\sigma[/ilmath]-algebra | The events we can have |
|
[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
- ↑ p22 - Measures, Integrals and Martingales - Rene L. Schilling
