Exponential Distribution

Definition

Definition 139 (Exponential Distribution (PDF))

\(X\) is a continuous random variable with an exponential distribution if the probability density function is given by:

(188)\[\begin{split} \pdf(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if } x \geq 0 \\ 0 & \text{otherwise} \end{cases} \end{split}\]

where \(\lambda > 0\) is the rate parameter (rate of decay).

Some conventions:

1. We write \(X \sim \exponential(\lambda)\) to indicate that \(X\) has an exponential distribution with rate parameter \(\lambda\).

Remark 55 (Exponential Distribution (PDF))

When \(\lambda = 0\), we have,

\[ \pdf(x) = \pdf(0) = \lambda e^{-\lambda 0} = \lambda e^0 = \lambda \]

This means that \(\pdf(0)\) will be more than 1 if \(\lambda > 0\) [Chan, 2021].

Definition 140 (Exponential Distribution (CDF))

If \(X\) is a continuous random variable with an exponential distribution with rate parameter \(\lambda\), then the CDF is given by integrating the PDF defined in Definition 139:

(189)\[\begin{split} \cdf(x) = \begin{cases} 0 & \text{if } x < 0 \\ 1 - e^{-\lambda x} & \text{if } x \geq 0 \end{cases} \end{split}\]

The PDF and CDF of two exponential distributions are shown below.

Show code cell source

Hide code cell source

import sys
from pathlib import Path
parent_dir = str(Path().resolve().parents[2])
sys.path.append(parent_dir)

import numpy as np
import scipy.stats as stats

from reproducibility.seed import seed_all
from omnivault.utils.probability_theory.plot import plot_continuous_pdf_and_cdf
seed_all()

# x = np.linspace(0, 10, 5000)
lambda_1, lambda_2 = 0.5, 2
scale_1, scale_2 = 1 / lambda_1, 1 / lambda_2
X1 = stats.expon(scale=scale_1)
X2 = stats.expon(scale=scale_2)

plot_continuous_pdf_and_cdf(X1, 0, 10, title="Exponential$(\lambda = 0.5)$")
plot_continuous_pdf_and_cdf(X2, 0, 10, title="Exponential$(\lambda = 2)$")

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<>:20: SyntaxWarning: "\l" is an invalid escape sequence. Such sequences will not work in the future. Did you mean "\\l"? A raw string is also an option.
<>:19: SyntaxWarning: "\l" is an invalid escape sequence. Such sequences will not work in the future. Did you mean "\\l"? A raw string is also an option.
<>:20: SyntaxWarning: "\l" is an invalid escape sequence. Such sequences will not work in the future. Did you mean "\\l"? A raw string is also an option.
/tmp/ipykernel_4981/2726575731.py:19: SyntaxWarning: "\l" is an invalid escape sequence. Such sequences will not work in the future. Did you mean "\\l"? A raw string is also an option.
  plot_continuous_pdf_and_cdf(X1, 0, 10, title="Exponential$(\lambda = 0.5)$")
/tmp/ipykernel_4981/2726575731.py:20: SyntaxWarning: "\l" is an invalid escape sequence. Such sequences will not work in the future. Did you mean "\\l"? A raw string is also an option.
  plot_continuous_pdf_and_cdf(X2, 0, 10, title="Exponential$(\lambda = 2)$")

Expectation and Variance

Theorem 33 (Expectation and Variance of Exponential Distribution)

If \(X\) is a continuous random variable with an exponential distribution with rate parameter \(\lambda\), then the expectation and variance are given by:

(190)\[ \expectation \lsq X \rsq = \frac{1}{\lambda} \qquad \text{and} \qquad \var \lsq X \rsq = \frac{1}{\lambda^2} \]

References and Further Readings

Further readings is a must since Professor Chan give many intuition on how Exponential distribution is used in real life. He also showed how Exponential distribution is derived from the Poisson distribution.

• Chan, Stanley H. “Chapter 4.5. Uniform and Exponential Random Variables.” In Introduction to Probability for Data Science, 205-211. Ann Arbor, Michigan: Michigan Publishing Services, 2021.

• Pishro-Nik, Hossein. “Chapter 4.2.2. Exponential Distribution” In Introduction to Probability, Statistics, and Random Processes, 249-252. Kappa Research, 2014.