Rate parameter
Time between events (X)
P(x>X)
P(x≤X)
Mean
Median
Variance
Standard deviation

# Exponential Distribution Calculator

By Bogna Haponiuk

This exponential distribution calculator can help you determine the probability of a certain amount of time elapsing between two consecutive events. This article presents you with the definition and some examples of exponential distribution, as well as with the exponential distribution formula and an example of applying it in real life.

## What is exponential probability distribution?

Exponential distribution describes the time between events happening according to the Poisson distribution. It means that the events occur independently and at a constant average rate.

Its key property is being memoryless. It means that, for example, the probability of an hour elapsing before the next bus comes to the bus stop is the same in the morning and in the evening. Also, the probability of a car engine breaking down during the next hour is the same during the first and hundredth hour of it running - we "forget" about the state of the car. Geometric distribution also has this property.

You can simulate the exponential probability distribution using the SMp(x) function. While more versatile, it has more parameters, what makes it a bit more difficult to apply.

## Exponential distribution examples

Some examples of cases, in which the exponential distribution can be used, include:

• Time between goals in a match;
• Time between two buses coming to a bus stop;
• Time between two consecutive customers in a grocery store;
• Time between failures of a machine;
• Distance between two car accidents along a highway.

## Exponential distribution formula

The main formulas used for analysis of exponential distribution let you find the probability of time between two events being lower or higher than x:

`P(x>X) = exp(-a*x)` `P(x≤X) = 1 - exp(-a*x)`

where:

• a is the rate parameter of the distribution. It is the reciprocal of the average number of events in a time interval;
• P(x>X) is the probability of x being higher than the indicated value;
• P(x≤X) is the probability of x being lower than the indicated value;
• exp denotes the exponential function.

You can also find other values, including:

• mean - it is equal to `μ = 1/a`
• median - it is equal to `m = ln(2)/a`
• variance - it is equal to `σ² = 1/a²`
• standard deviation - it is equal to `σ = √(1/a²)`

## How to use the exponential distribution calculator?

Let's assume you run a school cafeteria. The average number of clients is 15 students per hour. What is the probability that you will have to wait not more than 3 minutes for a client to appear? We will solve this problem step by step.

1. Determine your base time interval. In this case, it will be most practical to use 1 minute as a time interval.
2. Make sure that your rate parameter is expressed as 'per base time interval'. We need to convert 15 students per hour to 15 students per 60 minutes, or 1 student per 4 minutes. Hence, the rate parameter is a = 1/4.
3. Determine the value of x. In this case, x is equal to 3 minutes. We want to calculate the value of P(x≤3).
4. Input these values to the exponential distribution formula:

`P(x≤X) = 1 - exp(-a*x)`

`P(x≤3) = 1 - exp(-0.25*3) = 0.528`

1. The probability of waiting less than 3 minutes is equal to 0.528, or 52.8%. If you want, you can also calculate the mean time between clients, the media, variance and standard deviation according to the formulas above. Make sure to check the results with the exponential distribution calculator!
Bogna Haponiuk

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