Continuous Uniform Distribution

Definition

Definition 113 (Continuous Uniform Distribution (PDF))

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

(187)\[\begin{split} \pdf(x) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \end{split}\]

where \([a,b]\) is the interval on which \(X\) is defined.

Some conventions:

1. We write \(X \sim \uniform(a,b)\) to indicate that \(X\) has a continuous uniform distribution on \([a,b]\).

Definition 114 (Continuous Uniform Distribution (CDF))

If \(X\) is a continuous random variable with a continuous uniform distribution on \([a,b]\), then the CDF is given by integrating the PDF defined in Definition 113:

(188)\[\begin{split} \cdf(x) = \begin{cases} 0 & \text{if } x < a \\ \frac{x-a}{b-a} & \text{if } a \leq x \leq b \\ 1 & \text{if } x > b \end{cases} \end{split}\]

The PDF and CDF of two continuous uniform distributions are shown below.

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import warnings

warnings.filterwarnings("ignore")
import numpy as np
import matplotlib.pyplot as plt

from scipy import stats

fig, axes = plt.subplots(2, 2, figsize=(10, 10))

u = np.linspace(-1, 9, 5000) # random variable realizations
U1 = stats.uniform(0.2, 0.8)

axes[0, 0].plot(u, U1.pdf(u), "r-", lw=3, alpha=0.6, label="Uniform[0.2, 0.8]")
axes[0, 0].set_title("PDF of Uniform[0.2, 0.8]")
axes[0, 0].set_xlim(-0.1, 1.3)
axes[0, 0].set_ylim(0, 3)
axes[0, 0].set_xlabel("x")
axes[0, 0].set_ylabel("pdf(x)")

axes[0, 1].plot(u, U1.cdf(u), "r-", lw=3, alpha=0.6, label="Uniform[0.2, 0.8]")
axes[0, 1].set_title("CDF of Uniform[0.2, 0.8]")
axes[0, 1].set_xlim(-1, 10)
axes[0, 1].set_ylim(0, 1.1)
axes[0, 1].set_xlabel("x")
axes[0, 1].set_ylabel("cdf(x)")

U2 = stats.uniform(2, 6)

axes[1, 0].plot(u, U2.pdf(u), "r-", lw=3, alpha=0.6, label="Uniform[2, 8]")
axes[1, 0].set_title("PDF of Uniform[2, 8]")
axes[1, 0].set_xlim(1, 9)
axes[1, 0].set_ylim(0, 1.1)
axes[1, 0].set_xlabel("x")
axes[1, 0].set_ylabel("pdf(x)")

axes[1, 1].plot(u, U2.cdf(u), "r-", lw=3, alpha=0.6, label="Uniform[2, 8]")
axes[1, 1].set_title("CDF of Uniform[2, 8]")
axes[1, 1].set_xlim(1, 9)
axes[1, 1].set_ylim(0, 1.1)
axes[1, 1].set_xlabel("x")
axes[1, 1].set_ylabel("cdf(x)")


plt.show()

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

# x = np.linspace(-1, 9, 5000) # random variable realizations
U1 = stats.uniform(0.2, 0.8)
U2 = stats.uniform(2, 6)

plot_continuous_pdf_and_cdf(U1, -1, 9, title="Uniform$([0.2, 0.8])$", xlim=(-0.1, 1.3), ylim=(0, 2))
plot_continuous_pdf_and_cdf(U2, -1, 10, title="Uniform$([2, 6])$", xlim=(-1, 10), ylim=(0, 1.1))

Expectation and Variance

Theorem 28 (Expectation and Variance of Continuous Uniform Distribution)

If \(X\) is a continuous random variable with a continuous uniform distribution on \([a,b]\), then

(189)\[ \expectation \lsq X \rsq = \frac{a+b}{2} \qquad \text{and} \qquad \var \lsq X \rsq = \frac{(b-a)^2}{12} \]

Remark 42 (Intuition for Expectation and Variance of Continuous Uniform Distribution)

The expectation of a continuous uniform distribution is the midpoint of the interval on which the random variable is defined.

This should not be surprising, since the probability density function is constant over the interval, and the probability of any point in the interval is the same.

Let’s say we have \(X \sim \text{Uniform}(0, 10)\), then \(\expectation \lsq X \rsq = 5\).

References and Further Readings

• Chan, Stanley H. “Chapter 4.5. Uniform and Exponential Random Variables.” In Introduction to Probability for Data Science, 201-205. Ann Arbor, Michigan: Michigan Publishing Services, 2021.

• Pishro-Nik, Hossein. “Chapter 4.2.1. Uniform Distribution” In Introduction to Probability, Statistics, and Random Processes, 248-248. Kappa Research, 2014.