Home page for accesible maths 7 Models for continuous random variables 7 Models for continuous random variables 7.2 Exponential Distribution:

{\rm Exp}(\beta)

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7.1 Uniform Distribution: ${\rm U}(a,b)$

A continuous random variable for which all outcomes in a given range have equal chance of occurring is said to be uniformly distributed. Specifically, a random variable $X$ has a Uniform distribution over the interval $(a,b)$ if the pdf is given by

f_{X}(x)=\left\{\begin{array}[]{ll}\frac{1}{b-a}&a<x<b,\\ &\\ 0&\mbox{otherwise.}\end{array}\right.

We write $X\sim{\rm U}(a,b)$ . This pdf for two different sets of parameter values is illustrated on Figure 7.1.

Figure 7.1: The pdfs for two uniformly distributed random variables $X$ with different parameter values.

We find that for all $x$ and $x+c$ such that $a<x<x+c<b$

{\rm P}(x<X\leq x+c)=c/(b-a),

so the probability of $X$ falling in any interval of length $c$ in the range $(a,b)$ is the same for all $x$ , i.e. independent of the position $x$ and proportional to the interval length $c$ .

Possible examples of Uniform random variables are: completely random numbers, the time of births in a 24 hour period, and the times of goals in a football match (Or are they?).

Example 7.1.

Find the cdf, expected value and variance of the ${\rm U}(a,b)$ distribution.

Solution.

The cdf is given by

$\displaystyle F_{X}(x)$	$\displaystyle=$	$\displaystyle\int_{-\infty}^{x}f_{X}(s)\,ds$
	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\int_{-\infty}^{x}0\,ds&\mbox{ if }x% \leq a,\\ \int_{-\infty}^{a}0\,ds+\int_{a}^{x}\frac{1}{b-a}\,ds&\mbox{ if }a<x\leq b,\\ \int_{-\infty}^{a}0\,ds+\int_{a}^{x}\frac{1}{b-a}\,ds+\int_{b}^{x}0\,ds&\mbox{% if }b<x,\end{array}\right.$
	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}0&\mbox{ if }x\leq a,\\ \frac{x-a}{b-a}&\mbox{ if }a<x\leq b,\\ 1&\mbox{ if }b<x.\end{array}\right.$

The expectation is

{\rm E}(X)=\int_{-\infty}^{\infty}xf_{X}(x)\,dx=\int_{a}^{b}\frac{x}{b-a}\,dx=% \frac{b^{2}-a^{2}}{2(b-a)}=\frac{b+a}{2}.

To calculate the variance, we first calculate

{\rm E}(X^{2})=\int_{-\infty}^{\infty}x^{2}f_{X}(x)\,dx=\int_{a}^{b}\frac{x^{2% }}{b-a}\,dx=\frac{b^{3}-a^{3}}{3(b-a)}=\frac{b^{2}+ab+a^{2}}{3}.

So the variance is

{\rm Var}(X)={\rm E}(X^{2})-({\rm E}(X))^{2}=\frac{b^{2}+ab+a^{2}}{3}-\left(% \frac{b+a}{2}\right)^{2}=\frac{(b-a)^{2}}{12}.

These results seem logical as if all values in the interval $(a,b)$ are equally likely then the expected value should be the average of the endpoints. Similarly the wider the interval the more variable the outcomes, hence the larger variance.

Exercise 7.2.

Find the upper quartile of the random variable ${X\sim{\rm U}(3,5)}$ .

Solution.

Note that ${x_{0.75}}$ lies in ${(3,5)}$ .

$\displaystyle{\rm P}(X<x_{0.75})$	$\displaystyle=$	$\displaystyle 0.75$
$\displaystyle\frac{x_{0.75}-3}{5-3}$	$\displaystyle=$	$\displaystyle 0.75$
$\displaystyle x_{0.75}$	$\displaystyle=$	$\displaystyle 4.5.$

Exercise 7.3.

If $X\sim\mbox{Uniform}(0,10)$ , use R to calculate the probability that (a) $X<3$ , (b) $X>6$ , (c) $3<X<8$ , (d) $8<X<10$ , (e) $8<X<13$ .

 punif(3,min=0,max=10)
 1-punif(6,0,10)
 punif(8,0,10)-punif(3,0,10)
 punif(10,0,10)-punif(8,0,10)
 punif(13,0,10)-punif(8,0,10)

Exercise 7.4.

Suppose that you know the score in your favourite team’s football league match was 1-0. You video the game. Assuming a Uniform distribution for the time of goals (and no half-time gap), what is the expected waiting time until the goal? What is the probability the goal is in the first half of the match?

Solution.

Let ${X}$ model the time to the goal. If ${X\sim{\rm U}(0,90)}$ then ${{\rm E}(X)=45}$ mins and $F(45)=0.5$ .

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7.1 Uniform Distribution: U⁢(a,b)

Example 7.1.

Solution.

Exercise 7.2.

Solution.

Exercise 7.3.

Exercise 7.4.

Solution.

7.1 Uniform Distribution: ${\rm U}(a,b)$