Difference between revisions of "Random variable"
(Created page with "==Definition== A '''Random variable''' is a measurable map from a probability space to any measurable space {{Def...") |
m |
||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
==Definition== | ==Definition== | ||
A '''Random variable''' is a [[Measurable map|measurable map]] from a [[Probability space|probability space]] to any [[Measurable space|measurable space]] | A '''Random variable''' is a [[Measurable map|measurable map]] from a [[Probability space|probability space]] to any [[Measurable space|measurable space]] | ||
| + | |||
| + | Let {{M|(\Omega,\mathcal{A},\mathbb{P})}} be a [[Probability space|probability space]] and let {{M|X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U}) }} be a random variable | ||
| + | |||
| + | |||
| + | Then: | ||
| + | |||
| + | <math>X^{-1}(U\in\mathcal{U})\in\mathcal{A}</math>, but anything <math>\in\mathcal{A}</math> is {{M|\mathbb{P} }}-measurable! So we see: | ||
| + | |||
| + | <math>\mathbb{P}(X^{-1}(U\in\mathcal{U}))\in[0,1]</math> which we may often write as: <math>\mathbb{P}(X=U)</math> for simplicity (see [[Mathematicians are lazy]]) | ||
| + | |||
| + | ==Notation== | ||
| + | Often a measurable space that is the domain of the RV will be a probability space, given as <math>(\Omega,\mathcal{A},\mathbb{P})</math>, and we may write either: | ||
| + | * {{M|X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\mathcal{U}) }} | ||
| + | * {{M|X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U}) }} | ||
| + | |||
| + | With the understanding we write {{M|\mathbb{P} }} in the top one only because it is convenient to remind ourselves what probability measure we are using. | ||
| + | |||
| + | ==Pitfall== | ||
| + | Note that it is only guaranteed that <math>X^{-1}(U\in\mathcal{U})\in\mathcal{A}</math> but it is not guaranteed that <math>X(A\in\mathcal{A})\in\mathcal{U}</math>, it may sometimes be the case. | ||
| + | |||
| + | For example consider the trivial [[Sigma-algebra|{{sigma|algebra}}]] <math>\mathcal{U}=\{\emptyset,V\}</math> | ||
| + | |||
| + | '''However''' If you consider <math>X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\{\emptyset,V\})</math> then this is just the random variable "something happens" underneath it all, or if <math>V=\{2,\cdots,12\}</math> the event that the sum of the scores is <math>\ge 2</math>. | ||
| + | |||
| + | ==Example== | ||
| + | ===Discrete random variable=== | ||
| + | {{Begin Example}} | ||
| + | Recall the roll two die example from [[Probability space|probability spaces]], we will consider the RV {{M|X}} = the sum of the scores | ||
| + | {{Begin Example Body}} | ||
| + | Recall the die example from [[Probability space|probability spaces]] (which is restated less verbosely here), there: | ||
| + | {|class="wikitable" border="1" | ||
| + | |- | ||
| + | ! Component | ||
| + | ! Definition | ||
| + | |- | ||
| + | | {{M|\Omega}} | ||
| + | | <math>\Omega=\{(a,b)|\ a,b\in\mathbb{N},\ a,b\in[0,6]\}</math> | ||
| + | |- | ||
| + | | {{M|\mathcal{A} }} | ||
| + | | <math>\mathcal{A}=\mathcal{P}(\Omega)</math> | ||
| + | |- | ||
| + | | {{M|\mathbb{P} }} | ||
| + | | <math>\mathbb{P}(A) = \frac{1}{36}|A|</math> | ||
| + | |} | ||
| + | |||
| + | Let us define the '''Random variable''' that is the sum of the scores on the die, that is <math>X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))</math>. | ||
| + | |||
| + | It should be clear that <math>(\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))</math> is a [[Measurable space|measurable space]] however we need not consider a measure on it. | ||
| + | |||
| + | Writing {{M|X}} out explicitly is hard but there are two parts to it: | ||
| + | |||
| + | '''Warning - the first bullet point is a suspected claim''' | ||
| + | |||
| + | * We can look at what generates a space, we need only consider the single events really, that is to say: | ||
| + | *: <math>X(A\in\mathcal{A})\cup X(B\in\mathcal{A})=X(A\cup B\in\mathcal{A})</math>, so we need only look at {{M|X}} of the individual events | ||
| + | {{Todo|Prove this}} | ||
| + | * We can write it more explicitly as: | ||
| + | *: <math>X(A\in\mathcal{A})=\{a+b|(a,b)\in A\}</math> | ||
| + | {{End Example Body}} | ||
| + | {{End Example}} | ||
| + | |||
| + | ====Example of pitfall==== | ||
| + | Take <math>X:(\Omega,\mathcal{P}(\Omega),\mathbb{P})\rightarrow(V,\mathcal{U})</math>, if we define <math>\mathcal{U}=\{\emptyset,V\}</math> then clearly: | ||
| + | |||
| + | <math>X(\{(1,2)\})=\{3\}\notin\mathcal{U}</math>. Yet it is still measurable. | ||
| + | |||
| + | So an example! <math>\mathbb{P}(X^{-1}(\{5\}))=\mathbb{P}(X=5)=\mathbb{P}(\{(1,4),(4,1),(2,3),(3,2)\})=\frac{4}{36}=\frac{1}{9}</math> | ||
| + | |||
| + | |||
{{Definition|Measure Theory|Statistics}} | {{Definition|Measure Theory|Statistics}} | ||
Latest revision as of 14:41, 20 March 2015
Contents
Definition
A Random variable is a measurable map from a probability space to any measurable space
Let [ilmath](\Omega,\mathcal{A},\mathbb{P})[/ilmath] be a probability space and let [ilmath]X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U}) [/ilmath] be a random variable
Then:
[math]X^{-1}(U\in\mathcal{U})\in\mathcal{A}[/math], but anything [math]\in\mathcal{A}[/math] is [ilmath]\mathbb{P} [/ilmath]-measurable! So we see:
[math]\mathbb{P}(X^{-1}(U\in\mathcal{U}))\in[0,1][/math] which we may often write as: [math]\mathbb{P}(X=U)[/math] for simplicity (see Mathematicians are lazy)
Notation
Often a measurable space that is the domain of the RV will be a probability space, given as [math](\Omega,\mathcal{A},\mathbb{P})[/math], and we may write either:
- [ilmath]X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\mathcal{U}) [/ilmath]
- [ilmath]X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U}) [/ilmath]
With the understanding we write [ilmath]\mathbb{P} [/ilmath] in the top one only because it is convenient to remind ourselves what probability measure we are using.
Pitfall
Note that it is only guaranteed that [math]X^{-1}(U\in\mathcal{U})\in\mathcal{A}[/math] but it is not guaranteed that [math]X(A\in\mathcal{A})\in\mathcal{U}[/math], it may sometimes be the case.
For example consider the trivial [ilmath]\sigma[/ilmath]-algebra [math]\mathcal{U}=\{\emptyset,V\}[/math]
However If you consider [math]X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\{\emptyset,V\})[/math] then this is just the random variable "something happens" underneath it all, or if [math]V=\{2,\cdots,12\}[/math] the event that the sum of the scores is [math]\ge 2[/math].
Example
Discrete random variable
Recall the roll two die example from probability spaces, we will consider the RV [ilmath]X[/ilmath] = the sum of the scores
Recall the die example from probability spaces (which is restated less verbosely here), there:
| Component | Definition |
|---|---|
| [ilmath]\Omega[/ilmath] | [math]\Omega=\{(a,b)|\ a,b\in\mathbb{N},\ a,b\in[0,6]\}[/math] |
| [ilmath]\mathcal{A} [/ilmath] | [math]\mathcal{A}=\mathcal{P}(\Omega)[/math] |
| [ilmath]\mathbb{P} [/ilmath] | [math]\mathbb{P}(A) = \frac{1}{36}|A|[/math] |
Let us define the Random variable that is the sum of the scores on the die, that is [math]X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))[/math].
It should be clear that [math](\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))[/math] is a measurable space however we need not consider a measure on it.
Writing [ilmath]X[/ilmath] out explicitly is hard but there are two parts to it:
Warning - the first bullet point is a suspected claim
- We can look at what generates a space, we need only consider the single events really, that is to say:
- [math]X(A\in\mathcal{A})\cup X(B\in\mathcal{A})=X(A\cup B\in\mathcal{A})[/math], so we need only look at [ilmath]X[/ilmath] of the individual events
TODO: Prove this
- We can write it more explicitly as:
- [math]X(A\in\mathcal{A})=\{a+b|(a,b)\in A\}[/math]
Example of pitfall
Take [math]X:(\Omega,\mathcal{P}(\Omega),\mathbb{P})\rightarrow(V,\mathcal{U})[/math], if we define [math]\mathcal{U}=\{\emptyset,V\}[/math] then clearly:
[math]X(\{(1,2)\})=\{3\}\notin\mathcal{U}[/math]. Yet it is still measurable.
So an example! [math]\mathbb{P}(X^{-1}(\{5\}))=\mathbb{P}(X=5)=\mathbb{P}(\{(1,4),(4,1),(2,3),(3,2)\})=\frac{4}{36}=\frac{1}{9}[/math]
