The Exponential Distribution

0.1 The Exponential Distribution

0.1.1 Important Formulas

The Exponential Distribution

Probability density function of the Exponential Distribution

The probability density function (pdf) of an exponential distribution is \[ {\displaystyle f(x;\lambda )={\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}}\]

Here \(\lambda > 0\) is the parameter of the distribution, often called the rate parameter.

0.1.2 Worked Examples

Worked Examples

Claim amounts are modelled as an exponential random variable with mean $1,000.

Exercises

  1. Calculate the probability that one such claim amount is greater than $5,000.

  2. Calculate the probability that a claim amount is greater than $5,000 given that it is greater than $1,000.

Solution

Click here for demonstrated solution

0.1.4 Review Questions

Review Question 1

An average of five calls per hour are received by a machine repair department. Beginning the observation at any point in time, what is the probability that the first call for service will arrive within five minutes? Jobs are sent to a printer at an average of 5 jobs per hour.

  1. What is the expected time between jobs?
  2. What is the probability that the next job is sent within 6 minutes after the previous job?

Review Question 2

Assume that the time, denominated in minutes, between arrivals of customers at a particular bank is exponentially distributed with a rate parameter of 0.25.

  1. What is the mean duration between arrivals?
  2. Find the probability that the time between arrivals is greater than 5 minutes.
  3. Find the probability that the time between arrivals will be less than 2 minute.

Review Question 3

Suppose that customers arrive at a filling station at the rate of 3 per hour. Given that a customer has just arrived, the time is takes for the next customer to arrive is called a waiting time}. Let $ T$ be the symbol for the ``waiting timesโ€ variable.

Suppose that a customer has just arrived.

  1. What is the expected waiting time between customer arrivals?
  2. Compute \(E(T)\)
  3. What is the variance of waiting times? Compute \(\mbox{Var}(T)\).
  4. What is the probability that the next customer will arrive within the next fifteen minutes?
  5. What is the probability that no customers arrive in the next half hour?