Exponential Distributions
Introduction
In this lesson, you'll learn about the Exponential Distribution and the types of questions you can answer using this distribution.
Objectives
You will be able to:
- Define the parameters of the exponential distribution
- Identify use cases for the exponential distribution
What is the Exponential Distribution?
The Exponential Distribution describes the probability distribution of the amount of time it takes before an event occurs. In a way, it solves the inverse of the problem solved by the Poisson Distribution.
Recall that the Poisson Distribution lets you ask how likely any given number of events are over a set interval of time.
The Exponential Distribution lets you ask how likely the length of an interval of time is before an event occurs exactly once.
Sample Problem Statements for the Exponential Distribution
Let's look at some examples of the type of questions you can answer with knowledge of the Exponential Distribution:
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How long before a sensor in this factory breaks down?
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How long until the next earthquake happens?
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How long will the next customer interaction take?
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How long until the next person visits my website?
As you can see, any type of question that we can ask about the length of time before a random event happens is a question that falls under the Exponential Distribution.
Parameters & Formulas
As with the other distributions we've learned about, our goal is to discover the probability that our Random Variable, $X$ will turn out to be a specific value,
In order to figure this out, we need to know the Decay Parameter,
Decay Rate Formula
Once we know the decay rate, we can use the Probability Density Function to tell us the exact point probability for any length
The Probability Density Function allows us to answer questions such as "What is the probability that it takes exactly 4 minutes to ring up this customer?"
Since we are talking about a continuously-valued function, we'll also often want to make use of the Cumulative Distribution Function. This allows us to answer questions such as "what is the probability that it will take less than 4 minutes to ring up this customer?"
Note that we discussed the mean of this distribution above, but not the standard deviation. This distribution is somewhat unique in that the standard deviation,
Solving a Sample Problem
Let's use Python to solve a sample problem described above. Let's assume that the average customer interaction at a grocery store takes 4 minutes. What is the probability that ringing up the next customer takes exactly 3 minutes? What about the probability that it takes 3 minutes or less?
Let's write a few quick functions to calculate the PDF and CDF for us in the cell below. Then, we can use it to solve the problem above.
import numpy as np
def exp_pdf(mu, x):
decay_rate = 1 / mu
return decay_rate * np.exp(-decay_rate * x)
def exp_cdf(mu, x):
decay_rate = 1 / mu
return 1 - np.exp(-decay_rate * x)
print("Point probability for exactly 3 minutes: {:.4f}%".format(exp_pdf(4, 3) * 100))
print("Cumulative probability of 3 minutes or less: {:.4f}%".format(exp_cdf(4, 3) * 100))
Point probability for exactly 3 minutes: 11.8092%
Cumulative probability of 3 minutes or less: 52.7633%
Summary
In this lesson, we learned about the Exponential Distribution, and how we can use the Decay Rate to calculate both the Probability Density Function and the Cumulative Distribution Function!