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Table of contents
1.
Introduction
2.
Exponential Distribution
3.
Exponential Distribution Formula
4.
Mean, Variance, and Standard Deviation
5.
Applications of Exponential Distribution
6.
Problems
7.
FAQs
8.
Key Takeaways
Last Updated: Mar 27, 2024

Exponential Distribution

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Prerita Agarwal
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23 Jul, 2024 @ 01:30 PM

Introduction

In this article, we will learn exponential distribution, its use, the mean and variance, and a question based on exponential distribution to understand it better.

Exponential Distribution

The exponential distribution is a continuous probability distribution of time between some specific events in a Poisson Point Process. The Poisson process is a process in which the events happen continuously and independently at a constant average rate. This can be understood using an example:

Poisson: Number of cars passing a toll-gate in an hour

Exponential: Number of hours between the car arrivals.

We can see that the Poisson process concerns the number of events per unit time and the Exponential distribution involves the amount of time per occurrence of an event.

Conditions that must be followed for an Exponential distribution are:

  1. Events must occur at a constant rate
  2. Events must be independent of each other

From the above two requirements, we can conclude that the Exponential distribution is memory-less.

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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 λ is the distribution rate

The exponential distribution for different values of λ is given in the graph below.

Source

mean = 1/λ

variance = 1/λ2

Standard deviation = 1/λ

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/λ

Now, let's derive the variance of the exponential distribution.

Source

Therefore, variance = 1/λ2

We know that the standard deviation is the square root of the variance.

Therefore, standard deviation = 1/λ

Applications of Exponential Distribution

The exponential distribution is one of the most widely used continuous distributions. Some real-life applications of exponential distribution are given below.

  1. Predict the occurrence of an earthquake
  2. Call duration
  3. The life span of electronic gadgets
  4. Number of cars passing a traffic point per minute
  5. The time between customers arriving
  6. 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.

  1. Calculate the rate parameter
  2. Write the probability density function and graph it.
  3. What is the probability that a laptop will last less than three years?
  4. What is the probability that a laptop will last more than ten years?
  5. What probability will a laptop last between 4 and 7 years?

 

It is given that μ = 5years

Now, we know that μ = 1/λ

  1. λ(rate parameter) = ⅕ = 0.2
  2. PDF(Probability Density Function) = λ e-λx
    = 0.2 e-0.2x

Graph of the above PDF is given below

  1. P(X<x) = 1 - e-λx
    P(X<3) = 1 - e-0.2 x 3
    = 0.4512
    = 45.12 %

  1. P(X>x) = e-λx
    P(X>10) = e-0.2 x 10
    = 0.1353
    = 13.53 %

  1. 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

  1. What is exponential distribution?
    It is a continuous probability distribution between some specific events in a Poisson Process.
  2. 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.
  3. What is λ in exponential distribution?
    λ is the distribution rate in exponential distribution, and it defines the average number of events in an interval of time.
  4. What are the mean and variance of exponential distribution?
    Mean = 1/λ
    Variance = 1/λ2
  5. 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.

  1. Cauchy's Mean Value Theorem
  2. Mean, Variance, and Standard Deviation
  3. Rolle's Mean Value Theorem
  4. Poisson Distribution
  5. Prosecutor's fallacy

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