91Ó°ÊÓ

People are willing to commute to work is an exponential random variable with a decay parameter $\frac{1}{20}$

X = the distance people are willing to commute in miles.

The distance in miles is shown to be exponentially distributed with a decay value of $\frac{1}{20}$. It is known that the decay rate $\left(m\right)$is common of mean $\left(\mathrm{μ}\right)$for the exponential distribution. So,

$m=\frac{1}{20}$

Then,

$\mathrm{μ}=\frac{1}{m}$

$=\frac{1}{1/20}$

$=20$

It's also common knowledge that the mean of an exponential distribution equals its standard deviation. Hence,

$\mathrm{σ}=\mathrm{μ}$

$=20$

The exponential distribution's cumulative distribution function is now:

$P(X<x)=1−e^{−mx}$

As a result, the probability that a person will commute more than $25$miles can be computed as follows:

$P(x>25)=1−P(x<25)$

$=1−\left(1−e^{\frac{−1}{20}×25}\right)$

$=e^{−1.25}$

$=0.2865$