Let's go through the steps to solve this problem.

1. Part (a): Find $m$

   This question involves an exponential distribution, where the average (or mean) amount of time spent is given by $\mu = 18$ minutes.

   For an exponential distribution, the mean $\mu$ is related to the rate parameter $m$ by: $m = \frac{1}{\mu}$ Substituting $\mu = 18$: $m = \frac{1}{18}$

2. Part (b): Write the Probability Density Function $f(x)$

   The probability density function (PDF) for an exponential distribution with rate parameter $m$ is given by: $f(x) = m e^{-m x}$ Using $m = \frac{1}{18}$, we substitute this value into the PDF: $f(x) = \frac{1}{18} e^{-\frac{x}{18}}$

Thus, the answers are:

• Part (a): $m = \frac{1}{18}$
• Part (b): $f(x) = \frac{1}{18} e^{-\frac{x}{18}}$

Would you like further clarification on these steps?

Related Questions:

1. What is the variance of an exponential distribution, and how is it calculated?
2. How would you find the probability that a student takes more than 20 minutes using this distribution?
3. How does changing the average time $\mu$ affect the shape of the exponential distribution?
4. What are the properties of an exponential distribution that make it suitable for modeling time until an event occurs?
5. Can an exponential distribution have a negative mean value? Why or why not?

Tip:

For exponential distributions, remember that the mean and standard deviation are both equal to $\frac{1}{m}$, making it unique among common distributions.