Exponential Distribution Formula
If the Probability Density Function(PDF) of the continuous random variable is of the form as given below, then it is said to be an exponential distribution.
Where Î» is the distribution rate
The exponential distribution for different values of Î» is given in the graph below.
Source
mean = 1/Î»
variance = 1/Î»^{2}
Standard deviation = 1/Î»
Now, let's see how we got these values and derive these.
Mean, Variance, and Standard Deviation
The derivation of the mean of exponential distribution can be derived as:
Source
Therefore, mean = 1/Î»
Now, let's derive the variance of the exponential distribution.
Source
Therefore, variance = 1/Î»^{2}
We know that the standard deviation is the square root of the variance.
Therefore, standard deviation = 1/Î»
Applications of Exponential Distribution
The exponential distribution is one of the most widely used continuous distributions. Some reallife applications of exponential distribution are given below.
 Predict the occurrence of an earthquake
 Call duration
 The life span of electronic gadgets
 Number of cars passing a traffic point per minute
 The time between customers arriving
 The time that an interviewer spends with a candidate
We can see that exponential distribution is used in critical places like predicting earthquake occurrence. If you learned the exponential distribution, then congratulations, you just helped people evacuate from your building by warning them early of the earthquake.
Problems
Laptops produced by company XYZ last for five years on average. The life span of each laptop follows an exponential distribution.
 Calculate the rate parameter
 Write the probability density function and graph it.
 What is the probability that a laptop will last less than three years?
 What is the probability that a laptop will last more than ten years?
 What probability will a laptop last between 4 and 7 years?
It is given that Î¼ = 5years
Now, we know that Î¼ = 1/Î»
 Î»(rate parameter) = â…• = 0.2

PDF(Probability Density Function) = Î» e^{Î»x}
= 0.2 e^{0.2x}
Graph of the above PDF is given below

P(X<x) = 1  e^{Î»x}
P(X<3) = 1  e^{0.2 x 3}
= 0.4512
= 45.12 %

P(X>x) = e^{Î»x}
P(X>10) = e^{0.2 x 10}
= 0.1353
= 13.53 %

P(4<X<7) = P(X<7)  P(X<4)
= [1  e^{0.2 x 7}]  [1  e^{0.2 x 4}]
= 0.20273
= 20.273 %
FAQs

What is exponential distribution?
It is a continuous probability distribution between some specific events in a Poisson Process.

Differentiate between Poisson and exponential distribution.
Poisson distribution concerns the number of events per unit time, and Exponential distribution involves the amount of time per occurrence of an event.

What is Î» in exponential distribution?
Î» is the distribution rate in exponential distribution, and it defines the average number of events in an interval of time.

What are the mean and variance of exponential distribution?
Mean = 1/Î»
Variance = 1/Î»^{2}

Why is exponential distribution memoryless?
The exponential distribution is memoryless because events occur at a constant rate and are independent.
Key Takeaways
In this article, we have extensively discussed the exponential distribution topic. We hope that this blog has helped you enhance your engineering mathematics and statistics knowledge. If you would like to learn more, check out our other articles.
 Cauchy's Mean Value Theorem
 Mean, Variance, and Standard Deviation
 Rolle's Mean Value Theorem
 Poisson Distribution
 Prosecutor's fallacy
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