Suppose that the distance, in miles, that people are willing to commute to work is an exponential random variable with a decay parameter $\frac{1}{20}$. Let X = the distance people are willing to commute in miles. What is m, μ, and σ? What is the probability that a person is willing to commute more than 25 miles?

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 $\left(\mu \right)$for the exponential distribution. So,

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

Then,

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

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

$=20$

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

$\sigma =\mu$

$=20$

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:

$P(x>25)=1-P(x<25)$

$=1-\left(1-e^{\frac{-1}{20}\times 25}\right)$

$=e^{-1.25}$

$=0.2865$