Difference between revisions of "Random variable"

Revision as of 14:18, 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:

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

Example

Discrete random variable

[Expand]
Recall the roll two die example from probability spaces, we will consider the RV $X$ = the sum of the scores

Recall the die example from probability spaces (which is restated less verbosely here), there:

Component	Definition
$\Omega$	$\Omega=\{(a,b)\|\ a,b\in\mathbb{N},\ a,b\in[0,6]\}$
$\mathcal{A}$	$\mathcal{A}=\mathcal{P}(\Omega)$
$\mathbb{P}$	$\mathbb{P}(A) = \frac{1}{36}\|A\|$

Let us define the Random variable that is the sum of the scores on the die, that is $X:(\Omega,\mathcal{A},\mathbb{P})\rightarrow(\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))$ .

It should be clear that $(\{2,\cdots,12\},\mathcal{P}(\{2,\cdots,12\}))$ is a measurable space however we need not consider a measure on it.

Writing $X$ 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:
, so we need only look at $X$ of the individual events

TODO: Prove this

We can write it more explicitly as:
$X(A\in\mathcal{A})=\{a+b|(a,b)\in A\}$

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

@@ Line 25: / Line 25: @@
 ==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"
@@ Line 54: / Line 57: @@
 * 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====

Difference between revisions of "Random variable"

Revision as of 14:18, 20 March 2015

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Definition

Notation

Pitfall

Example

Discrete random variable

Example of pitfall

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