Random Variable

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  1. 1. Definition
  2. 2. A simple example

1.  Definition

Consider a probability space \fs4 (\Omega, \mathscr{F}, P). A random variable is an \fs4 \mathscr{F}-measurable function.

\fs4 X : \Omega \rightarrow \mathbb{R}^n

Essentially, the random variable is a quantification of an experiment.

2.  A simple example

Consider the following experiment, where we just toss a coin two times. The state space will be:

\fs4 \Omega = \{HH,HT,TH,TT\}

A \fs4 \sigma-algebra on this set can be the powerset (ie the set of all subsets). This the most extensive \fs4 \sigma-algebra that we can construct:

\fs4 \mathscr{F}_\infty = \{\emptyset, HH, HT, \ldots, \{HH,HT\}, \{HH,TH\},\ldots,\{HH,TT,TH\},\ldots,\Omega\}

Since \fs4 \mathscr{F}_\infty is a \fs4 \sigma-algebra, the pair \fs4 (\Omega, \mathscr{F}_\infty) is a measure space. Equipping this measure space with a probability measure \fs4 P defines a probability space \fs4 (\Omega, \mathscr{F}_\infty, P).

Note that we don't say the probability measure, but a probability measure, since this is not unique. For instance a biased coin will define a different probability space compared to a fair one.

Also remember that the probability measure attaches probabilities to members of \fs4 \mathscr{F}_\infty and not \fs4 \Omega. Therefore, the particular partition of the state space via \fs4 \mathscr{F}_\infty will serve as our information. In our example where we considered the powerset, the information is perfect.

An example of a random variable would be the number of heads tossed:

\fs4 X: \omega \rightarrow \mathbb{R} : X(\omega) = \{\text{# of }H\text{s in }\omega\}

Why is this function \fs4 \mathscr{F}_\infty-measurable? Because for any Borel set \fs4 B\in \mathscr{B}(\mathbb{R})

\fs4 X^{-1}(B) = \{\omega \in \Omega : X(\omega) \in B \} \in \mathscr{F}_\infty

For example, for \fs4 B=(3/4,6)

\fs4 X^{-1}((3/4,6)) = \{\omega \in \Omega : X(\omega) \in (3/4,6) \} = \{HT, TH, HH\} \in \mathscr{F}_\infty


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