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 battery continue to work before it dies?
- How long will a computer continue to work before it breaks down?
In each scenario, we’re interested in calculating how long we’ll have to wait until a certain event occurs. Thus, each scenario could be modeled using an exponential distribution.
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
To calculate probabilities related to the cumulative density function of the exponential distribution in Excel, we can use the following formula:
=EXPON.DIST(x, lambda, cumulative)
where:
- x: the value of the exponentially distributed random variable
- lambda: the rate parameter
- cumulative: whether to use the cumulative density function or not (TRUE or FALSE)
The following examples show how to use this formula in practice.
Example 1: Time Until Next Customer Arrives
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: The average time between customers is two minutes. Thus, the rate can be calculated as:
- λ = 1/μ
- λ = 1/2
- λ = 0.5
Thus, we can use the following formula in Excel to calculate the probability that a new customer arrives in less than one minute:
The probability that we’ll have to wait less than one minute for the next customer to arrive is 0.393469.
Example 2: Time Until Next Earthquake
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.
Solution: The average time between earthquakes is 400 days. Thus, the rate can be calculated as:
- λ = 1/μ
- λ = 1/400
- λ = 0.0025
Thus, we can use the following formula in Excel to calculate the probability that the next earthquake takes less than 500 days to occur:
The probability that it will take 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 3: Time Until Next Phone Call
Suppose 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: The average time between calls is 10 minutes. Thus, the rate can be calculated as:
- λ = 1/μ
- λ = 1/10
- λ = 0.1
Thus, we can use the following formula in Excel to calculate the probability that the next customer calls within 10 to 15 minutes:
The probability that a new customer calls within 10 to 15 minutes. is 0.1447.
Additional Resources
An Introduction to the Exponential Distribution
The Memoryless Property of the Exponential Distribution
How to Plot an Exponential Distribution in R