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 and at times it is referred to be a negative exponential 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}.
Calculation of the Exponential Distribution (Step by Step)
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.
 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
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.
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.
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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