*27*

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, 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.

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

*F*(x; Î») = 1 â€“ e^{-Î»x}

where:

**Î»:**the rate parameter (calculated as Î» = 1/Î¼)**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:

- Î» = 1/Î¼
- Î» = 1/2
- Î» = 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 weâ€™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:

- Î» = 1/Î¼
- Î» = 1/400
- Î» = 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 weâ€™ll have to wait *more* than 500 days for the next earthquake is 1 â€“ 0.7135 =Â **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:

- Î» = 1/Î¼
- Î» = 1/10
- Î» = 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