# 4 Real-Life Examples of 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.

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

F(x; λ) = 1 – e-λx

where:

• λ: the rate parameter (calculated as λ = 1/μ)
• e: A constant roughly equal to 2.718

In this article we share 5 examples of the exponential distribution in real life.

### Example 1: Time Between Geyser Eruptions

The number of minutes between eruptions for a certain geyser can be modeled by the exponential distribution.

For example, suppose the mean number of minutes between eruptions for a certain geyser is 40 minutes. If a geyser just erupts, what is the probability that we’ll have to wait less than 50 minutes for the next eruption?

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

• λ = 1/μ
• λ = 1/40
• λ = .025

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

• P(X ≤ x) = 1 – e-λx
• P(X ≤ 50) = 1 – e-.025(50)
• P(X ≤ 50) = 0.7135

The probability that we’ll have to wait less than 50 minutes for the next eruption is 0.7135.

### Example 2: Time Between Customers

The number of minutes between customers who enter a certain shop can be modeled by the exponential distribution.

For example, suppose 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.

To solve this, we can start by knowing that the average time between customers is two minutes. Thus, the rate can be calculated as:

• λ = 1/μ
• λ = 1/2
• λ = 0.5

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

• P(X ≤ x) = 1 – e-λx
• P(X ≤ 1) = 1 – e-0.5(1)
• P(X ≤ 1) = 0.3935

The probability that we’ll have to wait less than one minute for the next customer to arrive is 0.3935.

### Example 3: Time Between Earthquakes

The time between earthquake occurrences can be modeled using an exponential distribution.

For example, suppose 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.

To solve this, we start by knowing that the average time between earthquakes is 400 days. Thus, the rate can be calculated as:

• λ = 1/μ
• λ = 1/400
• λ = 0.0025

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

• P(X ≤ x) = 1 – e-λx
• P(X ≤ 1) = 1 – e-0.0025(500)
• P(X ≤ 1) = 0.7135

The probability that we’ll have to wait less than 500 days for the next earthquake is 0.7135.

Thus, the probability that we’ll have to wait more than 500 days for the next earthquake is 1 – 0.7135 = 0.2865.

### Example 4: Time Between Calls

The time between customer calls at different businesses can be modeled using an exponential distribution.

For example, suppose a bank 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.

To solve this , we start by knowing that the average time between calls is 10 minutes. Thus, the rate can be calculated as:

• λ = 1/μ
• λ = 1/10
• λ = 0.1

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

• P(10 < X ≤ 15) = (1 – e-0.1(15))  –  (1 – e-0.1(10))
• P(10 < X ≤ 15) = .7769 – .6321
• P(10 < X ≤ 15) = 0.1448

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