**Exponential Distribution (Table of Contents)**

## What is Exponential Distribution?

In statistics and probability theory, the expression of exponential distribution refers to the probability distribution that is used to define the time between two successive events that occur independently and continuously at a constant average rate. The exponential distribution is one of the extensively used continuous distributions and it is strictly related to the Poisson distribution.

### Exponential Distribution Formula

A continuous random variable *x* (with scale parameter λ > 0) is said to have an exponential distribution only if its probability density function can be expressed by multiplying the scale parameter to the exponential function of minus scale parameter and *x* for all *x* greater than or equal to zero, otherwise the probability density function is equal to zero.

Mathematically, the probability density function is represented as,

such that mean is equal to 1/ λ and variance is equal to 1/ λ^{2}.

### Explanation of the Exponential Distribution Formula

The formula for exponential distribution is derived by using the following steps:

**Step 1:** Firstly, try to figure out whether the event under consideration is continuous and independent in nature and occurs at a roughly constant rate. Any practical event will ensure that the variable is greater than or equal to zero.

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**Step 2:** Next, determine the value of the scale parameter, which is invariably the reciprocal of the mean.

**λ = 1 / mean**

**Step 3:** Next, multiply the scale parameter λ and the variable *x* and then calculate the exponential function of the product multiplied by minus one i.e. e^{– λ*x}.

**Step 4:** Finally, the probability density function is calculated by multiplying the exponential function and the scale parameter.

If the above formula holds true for all *x* greater than or equal to zero, then *x* is an exponential distribution.

### Example of Exponential Distribution

**Let us take the example of exponential distribution i.e x which is the amount of time taken (in minutes) by an office peon to deliver from the manager’s desk to the clerk’s desk. The function of time taken is assumed to have an exponential distribution with the average amount of time equal to five minutes.**

Given that *x* is a continuous random variable since time is measured.

Average, μ = 5 minutes

Therefore, scale parameter, λ = 1 / μ = 1 / 5 = 0.20

Hence, the exponential distribution probability function can be derived as,

**f(x) = 0.20 e ^{– 0.20*x}**

Now, calculate the exponential distribution probability function at different values of *x* to derive the distribution curve.

**For x = 0**

exponential distribution probability function for x=0 will be,

Similarly, calculate exponential distribution probability function for x=1 to x=30

- For x = 0, f(0) = 0.20 e
^{-0.20*0}= 0.200 - For x = 1, f(1) = 0.20 e
^{-0.20*1}= 0.164 - For x = 2, f(2) = 0.20 e
^{-0.20*2}= 0.134 - For x = 3, f(3) = 0.20 e
^{-0.20*3}= 0.110 - For x = 4, f(4) = 0.20 e
^{-0.20*4}= 0.090 - For x = 5, f(5) = 0.20 e
^{-0.20*5}= 0.074 - For x = 6, f(6) = 0.20 e
^{-0.20*6}= 0.060 - For x = 7, f(7) = 0.20 e
^{-0.20*7}= 0.049 - For x = 8, f(8) = 0.20 e
^{-0.20*8}= 0.040 - For x = 9, f(9) = 0.20 e
^{-0.20*9}= 0.033 - For x = 10, f(10) = 0.20 e
^{-0.20*10}= 0.027 - For x = 11, f(11) = 0.20 e
^{-0.20*11}= 0.022 - For x = 12, f(12) = 0.20 e
^{-0.20*12}= 0.018 - For x = 13, f(13) = 0.20 e
^{-0.20*13}= 0.015 - For x = 14, f(14) = 0.20 e
^{-0.20*14}= 0.012 - For x = 15, f(15) = 0.20 e
^{-0.20*15}= 0.010 - For x = 16, f(16) = 0.20 e
^{-0.20*16}= 0.008 - For x = 17, f(17) = 0.20 e
^{-0.20*17}= 0.007 - For x = 18, f(18) = 0.20 e
^{-0.20*18}= 0.005 - For x = 19, f(19) = 0.20 e
^{-0.20*19}= 0.004 - For x = 20, f(20) = 0.20 e
^{-0.20*20}= 0.004 - For x = 21, f(21) = 0.20 e
^{-0.20*21}= 0.003 - For x = 22, f(22) = 0.20 e
^{-0.20*22}= 0.002 - For x = 23, f(23) = 0.20 e
^{-0.20*23}= 0.002 - For x = 24, f(24) = 0.20 e
^{-0.20*24}= 0.002 - For x = 25, f(25) = 0.20 e
^{-0.20*25}= 0.001 - For x = 26, f(26) = 0.20 e
^{-0.20*26}= 0.001 - For x = 27, f(27) = 0.20 e
^{-0.20*27}= 0.001 - For x = 28, f(28) = 0.20 e
^{-0.20*28}= 0.001 - For x = 29, f(29) = 0.20 e
^{-0.20*29}= 0.001 - For x = 30, f(30) = 0.20 e
^{-0.20*30}= 0.000

We have derived distribution curve as follows,

### Relevance and Use

Although the assumption of a constant rate is very rarely satisfied in the real world scenarios, if the time interval is selected in such a way that the rate is roughly constant, then the exponential distribution can be used as a good approximate model. It has many other applications in the field of physics, hydrology, etc.

### Recommended Articles

This has been a guide to the Exponential Distribution. Here we discuss how to calculate exponential distribution using its formula along with an example and downloadable excel template. You can learn more about statistical modeling from the following articles –

- Formula of T Distribution
- Formula of Binomial Distribution
- Lognormal Excel Distribution
- Frequency Excel Distribution

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