The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes. Write the distribution, state the probability density function, and graph the distribution.

The amount of time spouses shop for anniversary cards can be modeled by an exponential distribution with the average amount of time equal to eight minutes.

It is shown that the amount of time spouses shop is exponentially distributed with mean 8 minutes. It is known that the decay rate $\left(m\right)$is reciprocal of mean $\left(\mu \right)$for the exponential distribution.

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

$=\frac{1}{8}$

$=0.125$

Thus the amount of time spouses shop is exponentially distributed is,

$X~\mathrm{Exp}\left(0.125\right)$

Where,

$X=\text{Random variable for the amount of time spouses shop}$

Again, the probability density function for this exponential distribution is calculated below,

$f\left(x\right)=m{e}^{-mx}$

Now give value to $m$,

$f\left(x\right)=0.125e^{-0.125x}$

Now, make the graph by using the above-obtained probability density function. The maximum value of $f\left(x\right)$ which will lie on the y-axis and at $x=0$will be:

$f\left(0\right)=0.125e^{0.125\times 0}$

$=0.125e^0$

$=0.125$

$=m$