Home page for accesible maths 4.1 Definition 4.1 Definition 4.2 Probability mass functions

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4.1.1 Introductory examples of random variables

Discrete random variables arise in a variety of ways: From experiments

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with a natural integer valued outcome
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  the number of buses to stop in the hour,
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  the number of goals in a football match.
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with a continuous outcome which is recorded on an integer scale
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  heights, ages, salaries
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with non-integer outcomes to which numerical values are assigned
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  a coin is tossed the outcome is ${H}$ or ${T}$ , converted to ${1}$ and ${0}$ respectively.
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  disease stage coded on a numerical scale (e.g. 1,2,…,5).

Exercise 4.2.

A coin is tossed 3 times. The sample space is

{\Omega=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}.}

Define a rv giving the number of ${H}$ s thrown.

Solution.

Define ${R(\omega)=\#H\mbox{ in }\omega}$ .

Long hand:

	$\displaystyle R(TTT)=0$
	$\displaystyle R(HTT)=R(THT)=R(TTH)=1$
	$\displaystyle R(HHT)=R(HTH)=R(THH)=2$
	$\displaystyle R(HHH)=3.$

The induced sample space for ${R}$ is ${{\mathcal{S}}=\{0,1,2,3\}}$ .

Example 4.3.

Suppose we decide to record the number of children born in the local maternity ward tomorrow as a probability experiment. Find a suitable sample space and random variable.

Solution.

Any outcome is a non-negative integers, so a suitable sample space is ${\Omega=\{0,1,2,\dots\}}$ .

The rv is ${R(\omega)=\omega}$ , giving the number of children born.

Suppose that ${A\subset{\mathcal{S}}}$ is an event in the induced sample space. Then we write

\displaystyle\{R\in A\}=\{\omega\in\Omega:R(\omega)\in A\}.

In the 3 coins example above, we have that

{\{R=2\}=\{HHT,HTH,THH\}}.

The right hand side of these equations is an event in ${\Omega}$ and hence has a probability assigned to it. This induces a probability on the induced sample space

\displaystyle{\rm P}(R\in A)=P(\{\omega\in\Omega:R(\omega)\in A\}).

Exercise 4.4.

Suppose our sample space consists of the outcomes of throwing a fair die, and suppose we gamble on the outcome:

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lose ${\pounds 1}$ if outcome is ${1}$ , ${2}$ or ${3}$ ;
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win nothing if outcome is ${4}$ ;
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win ${\pounds 2}$ if outcome is ${5}$ or ${6}$ .

Define ${R}$ to be the random variable giving the profit. Find the induced sample space for ${R}$ , and evaluate the probabilities on the induced sample space.

Solution.

${\Omega=\{1,2,3,4,5,6\}}$ and

	$\displaystyle R(1)=R(2)=R(3)=-1,$
	$\displaystyle R(4)=0,$
	$\displaystyle R(5)=R(6)=2.$

The induced sample space for ${R}$ is ${{\mathcal{S}}=\{-1,0,2\}}$ . Now

$\displaystyle{\rm P}(R=-1)$	$\displaystyle=$	$\displaystyle{\rm P}(\{1,2,3\})=\frac{1}{2},$
$\displaystyle{\rm P}(R=0)$	$\displaystyle=$	$\displaystyle{\rm P}(\{4\})=\frac{1}{6},$
$\displaystyle{\rm P}(R=2)$	$\displaystyle=$	$\displaystyle{\rm P}(\{5,6\})=\frac{1}{3}.$

The probability associated to the other subsets can be obtained using axiom 3 e.g.

\displaystyle{\rm P}(R\in\{-1,0\})={\rm P}(R=-1)+{\rm P}(R=0)=\frac{1}{2}+% \frac{1}{6}=\frac{2}{3}.

To summarise

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The outcomes in the sample space, ${\Omega}$ , of the probability experiment may or may not be numerically valued.
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A random variable ${R}$ is a function that associates a unique real number with each outcome in the sample space, ${\Omega}$ .
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A random variable is not a number. It is neither random, nor a variable. It is a function.
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The set of values taken by random variable ${R}$ defined on ${\Omega}$ , is known as the induced sample space for ${R}$ and is sometimes written as ${{\mathcal{S}}}$ .
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The event ${\{R=r\}=\{\omega:\;R(\omega)=r\}}$ and this equivalence induces a probability distribution on ${{\mathcal{S}}}$ .