# How to use the EXPON.DIST function

**What is the EXPON.DIST function?**

The EXPON.DIST function calculates the exponential distribution representing an outcome in the form of probability.

This function was introduced in Excel 2010, it has replaced the EXPONDIST function which is now outdated.

### Table of Contents

## 1. Introduction

**What is an exponential distribution?**

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson point process. It is characterized by a constant hazard rate so that the conditional probability of an event does not depend on how much time has passed already.

**What is a hazard rate?**

The hazard rate is the conditional probability that an event will occur in a small interval given it has not yet occurred. It is often assumed constant in Poisson processes and given by the exponential distribution's parameter.

**What is a Poisson point process?**

A Poisson point process is a random collection of points representing events in time or space that follow a Poisson distribution. It has independence between points and a constant rate of average occurrence over any interval.

**What is a Poisson distribution?**

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space. The events must be independent and their average rate is known and constant.

**What is independence between points?**

Independence between points means that the position or timing of one point in a stochastic spatial or temporal point process does not affect or influence the probability distribution governing the other points. In other words, it is completely random.

**What is a stochastic spatial process?**

A stochastic spatial process is a random mechanism for generating points distributed in space according to some probability distribution and spatial relationship between the points.

**What is a temporal point process?**

A temporal point process generates random events in time such as the arrival of customers or radioactive decay with certain probabilistic properties concerning the timing and intervals between events.

**What is memorylessness?**

Memorylessness is the property of certain probability distributions like the exponential and geometric where past outcomes do not affect future ones, so the conditional probability stays constant regardless of history.

**What is a probability distribution?**

A probability distribution lets you analyze how likely different random values occurs, in other words, it shows how often we would expect to see different potential values.

**What is a continuous probability distribution?**

A continuous probability distribution is defined over an interval and range of continuous values, giving the probability an outcome is exactly equal to any value, and having an area under its probability density curve equal to 1.

## 2. EXPON.DIST function Syntax

EXPON.DIST(*x*, *lambda*, *cumulative*)

## 3. EXPON.DIST function Arguments

x |
Required. The value of the function. |

lambda |
Required. The parameter value. |

cumulative |
Required. A boolean value. TRUE - cumulative distribution function. FALSE - probability density function. |

## 4. EXPON.DIST function Example 1

**In a manufacturing plant, the time between failures of a certain machine follows an exponential distribution with a mean of 2 hours. What is the probability that the time until the next failure is less than 1.5 hours?**

The arguments in the EXPON.DIST function are:

- x = 1.5
- lambda = 1/2 = 0.5
- cumulative = true

The image above shows the arguments in cells C18, C19, and C20. The calculation output is in cell C21. The chart above shows the exponential density function as a blue line and the cumulative exponential density function as an orange line.

Formula in cell E3:

The probability that the time until next machine failure is less than 1.5 hours or 90 minutes is 0.5276 or 52.76%

Find the x value 1.5 on the chart above, go straight up until you cross the orange line. Now go straight left to the y-axis and the value you will find is approx 0.53

## 5. EXPON.DIST function Example 2

**In a call center, the time between incoming calls follows an exponential distribution with a mean of 3 minutes. What is the probability that the time until the next call is more than 5 minutes?**

The arguments in the EXPON.DIST function are:

- x = 5
- lambda = 1/3
- cumulative = true

The image above shows the arguments in cells C18, C19, and C20. The calculation output is in cell C21. The chart above shows the exponential density function as a blue line and the cumulative exponential density function as an orange line.

Formula in cell E3:

The probability that the time until next call is more than 5 minutes is the complement to 0.811 which is 1 - 0.811 = 0.1889 or 18.89%

Find the x value 5 on the x-axis on the chart above, go straight up until you cross the orange line. Now go straight left to the y-axis and the value you will find is approx 0.19

## 6. EXPON.DIST function Example 3

**On the average, a certain car model lasts 20 years. The length of time the car model lasts is exponentially distributed. What is the probability that the car model lasts more than 15 years?**

The arguments in the EXPON.DIST function are:

- x = 15
- lambda = 1/20
- cumulative = true

The image above shows the arguments in cells C18, C19, and C20. The calculation output is in cell C21. The chart above shows the exponential density function as a blue line and the cumulative exponential density function as an orange line.

Formula in cell E3:

The probability that the car lasts more than 15 years is 0.5276 or 52.76%

Find the x value 15 on the x-axis on the chart above, go straight up until you cross the orange line. Now go straight left to the y-axis and the value you will find is approx 0.5276

## 7. EXPON.DIST function not working

The EXPON.DIST function returns

- #VALUE! error value if
*lambda*or*x*is non-numeric. - #NUM! error value if:
- x < 0 (zero)
- lambda <= 0 (zero)

## 8. How is the EXPON.DIST function calculated?

The general equation to calculate the cumulative distribution function:

F(x;λ) = 1-e^{-λx}

e - exp

λ - rate parameter

x - random variable

The general equation to calculate the probability density distribution:

F(x;λ) = λe^{-λx}

Link to Wikipedia: Exponential distribution

### Functions in 'Statistical' category

The EXPON.DIST function function is one of 73 functions in the 'Statistical' category.

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