Home page for accesible maths 7 Models for continuous random variables 7.1 Uniform Distribution:

{\rm U}(a,b)

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

7.2 Exponential Distribution: ${\rm Exp}(\beta)$

A random variable $X$ has an exponential distribution with rate $\beta$ if its pdf is given by

f_{X}(x)=\left\{\begin{array}[]{ll}C\exp(-\beta x)&x\geq 0,\\ 0&\mbox{otherwise,}\end{array}\right.

where $\beta>0$ and $C$ is a normalising constant. We write $X\sim{\rm Exp}(\beta)$ .

Exercise 7.5.

For which value of $C$ is this a valid pdf (i.e. is non-negative and integrates to 1)?

Solution.

For $f_{X}(x)$ to be non-negative, we need $C\geq 0$ .
We also need

1=\int_{-\infty}^{\infty}f_{X}(x)=\int_{0}^{\infty}C\exp(-\beta x)\,dx=\left[% \frac{C}{-\beta}\exp(-\beta x)\right]_{0}^{\infty}=\frac{C}{\beta}.

This means that $C=\beta$ .

The pdf for two different values of $\beta$ is shown in Figure 7.2.

Figure 7.2: The pdfs for two exponentially-distributed random variables $X$ with $\beta=1$ and $\beta=2$ .

Exercise 7.6.

Find the cdf of the Exponential $(\beta)$ distribution.

Solution.

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

Example 7.7.

The Exponential distribution is often used to model waiting times. Suppose that the length of a phone call to a company, in minutes, is distributed as ${\rm Exp}(1/10)$ . If you phone and the number is busy then you are put on hold until the operator is free. If someone phones the company immediately before you, find the probability that you will have to wait (a) less than 10 minutes, (b) between 10 and 20 minutes, and (c) between 10 and 20 minutes once you have already waited for 10 minutes.

Solution.

Let ${X}$ be the time you have to wait, which equals the length of the call. $X\sim{\rm Exp}(1/10)$ .

a.

${{\rm P}(X<10)=F_{X}(10)=1-\exp(-\frac{1}{10}10)=1-e^{-1}}$ .
b.

${\rm P}(10<X<20)={F_{X}(20)-F_{X}(10)=e^{-1}-e^{-2}}$ .
c.

$\displaystyle{\rm P}(10<X<20|X>10)$ $\displaystyle=$ $\displaystyle{\rm P}(10<X<20\cap X>10)/{\rm P}(X>10)$

$\displaystyle=$ $\displaystyle{\rm P}(10<X<20)/{\rm P}(X>10)$

$\displaystyle=$ $\displaystyle\frac{e^{-1}-e^{-2}}{e^{-1}}=1-e^{-1}.$

7.2.1 Lack of memory

7.2.2 The Gamma function

7.2.3 Expectation and variance of ${\rm Exp}(\beta)$ rvs

$\displaystyle{\rm P}(10<X<20\|X>10)$	$\displaystyle=$	$\displaystyle{\rm P}(10<X<20\cap X>10)/{\rm P}(X>10)$
	$\displaystyle=$	$\displaystyle{\rm P}(10<X<20)/{\rm P}(X>10)$
	$\displaystyle=$	$\displaystyle\frac{e^{-1}-e^{-2}}{e^{-1}}=1-e^{-1}.$

Style control - access keys in brackets

7.2 Exponential Distribution: Exp⁢(β)

Exercise 7.5.

Solution.

Exercise 7.6.

Solution.

Example 7.7.

Solution.

7.2 Exponential Distribution: ${\rm Exp}(\beta)$