Difference between revisions of "Random variable"

Revision as of 09:02, 19 March 2015 (view source) Alec (Talk | contribs) m ← Older edit		Latest revision as of 14:41, 20 March 2015 (view source) Alec (Talk | contribs) m
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	For example consider the trivial [[Sigma-algebra|{{sigma|algebra}}]] <math>\mathcal{U}=\{\emptyset,V\}</math>		For example consider the trivial [[Sigma-algebra|{{sigma|algebra}}]] <math>\mathcal{U}=\{\emptyset,V\}</math>
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		+	'''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==		==Example==
	===Discrete random variable===		===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:		Recall the die example from [[Probability space|probability spaces]] (which is restated less verbosely here), there:
	{|class="wikitable" border="1"		{|class="wikitable" border="1"
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	* We can write it more explicitly as:		* We can write it more explicitly as:
	*: <math>X(A\in\mathcal{A})=\{a+b|(a,b)\in A\}</math>		*: <math>X(A\in\mathcal{A})=\{a+b|(a,b)\in A\}</math>
		+	{{End Example Body}}
		+	{{End Example}}
	====Example of pitfall====		====Example of pitfall====

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Latest revision as of 14:41, 20 March 2015

Definition

A Random variable is a measurable map from a probability space to any measurable space

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and let $X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U})$ be a random variable

Then:

$$X^{-1}(U\in\mathcal{U})\in\mathcal{A}$$ , but anything $$\in\mathcal{A}$$ is $\mathbb{P}$-measurable! So we see:

$$\mathbb{P}(X^{-1}(U\in\mathcal{U}))\in[0,1]$$ which we may often write as: $$\mathbb{P}(X=U)$$ 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

$$(\Omega,\mathcal{A},\mathbb{P})$$ , and we may write either:

• $X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\mathcal{U})$
• $X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U})$

With the understanding we write $\mathbb{P}$ 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

$$X^{-1}(U\in\mathcal{U})\in\mathcal{A}$$ but it is not guaranteed that $$X(A\in\mathcal{A})\in\mathcal{U}$$ , it may sometimes be the case.

For example consider the trivial $\sigma$-algebra

$$\mathcal{U}=\{\emptyset,V\}$$

However If you consider

$$X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(V,\{\emptyset,V\})$$ then this is just the random variable "something happens" underneath it all, or if $$V=\{2,\cdots,12\}$$ the event that the sum of the scores is $$\ge 2$$ .

Example

Discrete random variable

Example of pitfall

Take

$$X:(\Omega,\mathcal{P}(\Omega),\mathbb{P})\rightarrow(V,\mathcal{U})$$ , if we define $$\mathcal{U}=\{\emptyset,V\}$$ then clearly:

$$X(\{(1,2)\})=\{3\}\notin\mathcal{U}$$ . Yet it is still measurable.

So an example!

$$\mathbb{P}(X^{-1}(\{5\}))=\mathbb{P}(X=5)=\mathbb{P}(\{(1,4),(4,1),(2,3),(3,2)\})=\frac{4}{36}=\frac{1}{9}$$