Exponential Distribution

What is Exponential Distribution?

The exponential distribution refers to the continuous and constant probability distribution which is actually used to model the time period that a person needs to wait before the given event happens and this distribution is a continuous counterpart of a geometric distribution that is instead distinct.

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,

exponential distribution formula
Exponential-Distribution-Formula

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For eg:
Source: Exponential Distribution (wallstreetmojo.com)

such that mean is equal to 1/ λ, and variance is equal to 1/ λ2.

Calculation of the Exponential Distribution (Step by Step)

Follow the below steps:

  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.

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

    λ = 1 / mean

  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.

  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

You can download this Exponential Distribution Excel Template here – Exponential Distribution Excel Template

Let us take the example, 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

ED Formula example 1.1

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

f(x) = 0.20 e– 0.20*x

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

For x = 0

ED Formula example 1.2

exponential distribution probability function for x=0 will be,

Exponential Distribution Formula example 1.3

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,

ED Formula example 1.4

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.

In statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more 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. It is one of the extensively used continuous distributions and it is strictly related to the Poisson distribution in excelPoisson Distribution In ExcelThe Poisson Distribution is a type of distribution that is used to calculate the frequency of events that will occur at any given time but are independent of one another.read more.

Recommended Articles

This article 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 –

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