Home page for accesible maths 5 Models for discrete random variables Binomial expansion 5.3 Bernoulli random variables

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5.2 Discrete uniform random variables

Consider an experiment where the sample space is ${\{0,1,\ldots,m\}}$ and the random variable ${R}$ corresponds to an outcome being picked at random from the sample space. Each outcome is equi-probable. Examples include:

•

the score on a die with faces enumerated 0,…,5,
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the number of heads on the toss of a fair coin,
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the day of the year of a randomly selected person’s birthday (days numbered from 0 to 364)

Exercise 5.1.

Write down the pmf of a discrete uniform rv.

Solution.

For $r=0,1,\ldots,m$ ,

p_{R}(r)={\color[rgb]{1,1,1}{\frac{1}{m+1}}}.

Otherwise $p_{R}(r)={\color[rgb]{1,1,1}{0.}}$

Example 5.2.

Calculate the expectation and variance of a discrete uniform random variable.

Solution.

$\displaystyle{\rm E}[R]$	$\displaystyle=$	$\displaystyle\sum_{r=0}^{\infty}r\,p_{R}(r)$
	$\displaystyle=$	$\displaystyle\sum_{r=0}^{m}r\,\frac{1}{m+1}$
	$\displaystyle=$	$\displaystyle\frac{1}{m+1}\sum_{r=0}^{m}r$
	$\displaystyle=$	$\displaystyle\frac{1}{m+1}\;\frac{1}{2}m(m+1)=\frac{m}{2}.$

To find the variance we need ${\rm E}(R^{2})$ :

$\displaystyle{\rm E}[R^{2}]$	$\displaystyle=$	$\displaystyle\sum_{r=0}^{m}r^{2}\,\frac{1}{m+1}$
	$\displaystyle=$	$\displaystyle\frac{1}{m+1}\sum_{r=0}^{m}r^{2}$
	$\displaystyle=$	$\displaystyle\frac{1}{m+1}\;\frac{1}{6}m(m+1)(2m+1)$
	$\displaystyle=$	$\displaystyle\frac{m(2m+1)}{6}.$

Now

$\displaystyle{\rm Var}(R)$	$\displaystyle=$	$\displaystyle{\rm E}[R^{2}]-({\rm E}[R])^{2}$
	$\displaystyle=$	$\displaystyle\frac{m(2m+1)}{6}-\frac{m^{2}}{4}$
	$\displaystyle=$	$\displaystyle\frac{m(m+2)}{12}.$