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# An Introduction to the Exponential Distribution

The exponential distribution is a probability distribution that is used to model the time we must wait until a certain event occurs.

This distribution can be used to answer questions like:

• How long does a shop owner need to wait until a customer enters his shop?
• How long will a laptop continue to work before it breaks down?
• How long will a car battery continue to work before it dies?
• How long do we need to wait until the next volcanic eruption in a certain region?

In each scenario, weâ€™re interested in calculating how long weâ€™ll have to wait until a certain event occurs. Thus, each scenario could be modeled using an exponential distribution.

### Exponential Distribution: PDF & CDF

If a random variable X follows an exponential distribution, then the probability density function ofÂ X can be written as:

f(x; Î») = Î»e-Î»x

where:

• Î»: the rate parameter (calculated as Î» = 1/Î¼)
• e: A constant roughly equal to 2.718

TheÂ cumulative distribution function ofÂ X can be written as:

F(x; Î») = 1 â€“ e-Î»x

In practice, the CDF is used most often to calculate probabilities related to the exponential distribution.

For example, suppose the mean number of minutes between eruptions for a certain geyser is 40 minutes. What is the probability that weâ€™ll have to wait less than 50 minutes for an eruption?

To solve this, we need to first calculate the rate parameter:

• Î» = 1/Î¼
• Î» = 1/40
• Î» = .025

We can plug in Î» = .025 and x = 50 to the formula for the CDF:

• P(X â‰¤ x) = 1 â€“ e-Î»x
• P(X â‰¤ 50) = 1 â€“ e-.025(50)
• P(X â‰¤ 50) = 0.7135

The probability that weâ€™ll have to wait less than 50 minutes for the next eruption isÂ 0.7135.

### Visualizing the Exponential Distribution

The following plot shows the probability density functionÂ of a random variableÂ X that follows an exponential distribution with different rate parameters:

And the following plot shows the cumulative distribution function of a random variable X that follows an exponential distribution with different rate parameters:

Note: Check out this tutorial to learn how to plot an exponential distribution in R.

### Properties of the Exponential Distribution

The exponential distribution has the following properties:

• Mean: 1 / Î»
• Variance: 1 / Î»2

For example, suppose the mean number of minutes between eruptions for a certain geyser is 40 minutes. We would calculate the rate as Î» = 1/Î¼ = 1/40 = .025.

We could then calculate the following properties for this distribution:

• Mean waiting time for next eruption: 1/Î» = 1 /.025 = 40
• Variance in waiting times for next eruption: 1/Î»2 = 1 /.0252 = 1600

Note: The exponential distribution also has a memoryless property, which means the probability of some future event occurring is not affected by the occurrence ofÂ  past events.

### Exponential Distribution Practice Problems

Use the following practice problems to test your knowledge of the exponential distribution.

Question 1: A new customer enters a shop every two minutes, on average. After a customer arrives, find the probability that a new customer arrives in less than one minute.

Solution 1: The average time between customers is two minutes. Thus, the rate can be calculated as:

• Î» = 1/Î¼
• Î» = 1/2
• Î» = 0.5

We can plug in Î» = 0.5 and x = 1 to the formula for the CDF:

• P(X â‰¤ x) = 1 â€“ e-Î»x
• P(X â‰¤ 1) = 1 â€“ e-0.5(1)
• P(X â‰¤ 1) = 0.3935

The probability that weâ€™ll have to wait less than one minute for the next customer to arrive is 0.3935.

Question 2: An earthquake occurs every 400 days in a certain region, on average. After an earthquake occurs, find the probability that it will take more than 500 days for the next earthquake to occur.

Solution 2: The average time between earthquakes is 400 days. Thus, the rate can be calculated as:

• Î» = 1/Î¼
• Î» = 1/400
• Î» = 0.0025

We can plug in Î» = 0.0025 and x = 500 to the formula for the CDF:

• P(X â‰¤ x) = 1 â€“ e-Î»x
• P(X â‰¤ 1) = 1 â€“ e-0.0025(500)
• P(X â‰¤ 1) = 0.7135

The probability that weâ€™ll have to wait less than 500 days for the next earthquake is 0.7135. Thus, the probability that weâ€™ll have to waitÂ more than 500 days for the next earthquake is 1 â€“ 0.7135 =Â 0.2865.

Question 3: A call center receives a new call every 10 minutes, on average. After a customer calls, find the probability that a new customer calls within 10 to 15 minutes.

Solution 3: The average time between calls is 10 minutes. Thus, the rate can be calculated as:

• Î» = 1/Î¼
• Î» = 1/10
• Î» = 0.1

We can use the following formula to calculate the probability that a new customer calls within 10 to 15 minutes:

• P(10 -0.1(15))Â  â€“Â  (1 â€“ e-0.1(10))
• P(10
• P(10

The probability that a new customer calls within 10 to 15 minutes. is 0.1448.

### Additional Resources

The following tutorials provide introductions to other common probability distributions.