Uniform Distribution¶

Overview¶

The uniform distribution assigns equal probability to all values in an interval \([a, b]\). It is the simplest continuous distribution and serves as the foundation for random number generation, simulation, and probability integral transforms.

Definition¶

A random variable \(X\) follows a continuous uniform distribution on \([a, b]\):

\[ X \sim \text{Uniform}(a, b), \qquad f(x) = \begin{cases} \frac{1}{b - a} & \text{if } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases} \]

The PDF is constant over the interval, reflecting equal likelihood for all values.

CDF¶

\[ F(x) = \begin{cases} 0 & x < a \\ \frac{x - a}{b - a} & a \leq x \leq b \\ 1 & x > b \end{cases} \]

Properties¶

\[ \begin{aligned} E[X] &= \frac{a + b}{2} \\[4pt] \text{Var}(X) &= \frac{(b - a)^2}{12} \\[4pt] \text{SD}(X) &= \frac{b - a}{2\sqrt{3}} \end{aligned} \]

Derivation of Mean¶

\[ E[X] = \int_a^b x \cdot \frac{1}{b-a}\,dx = \frac{1}{b-a} \cdot \frac{x^2}{2}\bigg|_a^b = \frac{b^2 - a^2}{2(b-a)} = \frac{a+b}{2} \]

Derivation of Variance¶

\[ E[X^2] = \int_a^b x^2 \cdot \frac{1}{b-a}\,dx = \frac{1}{b-a} \cdot \frac{x^3}{3}\bigg|_a^b = \frac{a^2 + ab + b^2}{3} \]

\[ \text{Var}(X) = E[X^2] - (E[X])^2 = \frac{a^2 + ab + b^2}{3} - \frac{(a+b)^2}{4} = \frac{(b-a)^2}{12} \]

Standard Uniform Distribution¶

The special case \(U \sim \text{Uniform}(0, 1)\) is the standard uniform distribution. Any uniform variable can be related to it:

\[ X = a + (b - a)U \sim \text{Uniform}(a, b) \quad \text{where } U \sim \text{Uniform}(0, 1) \]

Conversely:

\[ U = \frac{X - a}{b - a} \sim \text{Uniform}(0, 1) \quad \text{where } X \sim \text{Uniform}(a, b) \]

Probability Integral Transform¶

The uniform distribution plays a central role in simulation through the probability integral transform:

Theorem: If \(X\) is a continuous random variable with CDF \(F\), then \(F(X) \sim \text{Uniform}(0, 1)\).

Converse (Inverse Transform Sampling): If \(U \sim \text{Uniform}(0, 1)\), then \(X = F^{-1}(U)\) has CDF \(F\).

Proof¶

\[ P(F(X) \leq u) = P(X \leq F^{-1}(u)) = F(F^{-1}(u)) = u \]

which is the CDF of \(\text{Uniform}(0, 1)\).

This theorem is the basis for generating random samples from any distribution using only a uniform random number generator.

Discrete Uniform Distribution¶

The discrete counterpart assigns equal probability to a finite set of values \(\{a, a+1, \ldots, b\}\):

\[ P(X = k) = \frac{1}{b - a + 1}, \quad k = a, a+1, \ldots, b \]

\[ E[X] = \frac{a + b}{2}, \qquad \text{Var}(X) = \frac{(b - a + 1)^2 - 1}{12} \]

Worked Example¶

Problem: Daily returns of a certain asset are modeled as uniformly distributed between \(-2\%\) and \(+3\%\). What is the probability the return exceeds \(1\%\)? What is the expected return?

Solution:

\[ P(X > 1) = \frac{3 - 1}{3 - (-2)} = \frac{2}{5} = 0.40 \]

\[ E[X] = \frac{-2 + 3}{2} = 0.5\% \]

Python: PDF, CDF, and Sampling¶

PDF and CDF¶

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

a, b = 2, 8
x = np.linspace(a - 1, b + 1, 300)

fig, ax = plt.subplots(figsize=(12, 3))
ax.plot(x, stats.uniform(loc=a, scale=b-a).pdf(x), label='PDF', lw=2)
ax.plot(x, stats.uniform(loc=a, scale=b-a).cdf(x), label='CDF', lw=2)
ax.spines[['top', 'right']].set_visible(False)
ax.legend()
plt.show()

Sampling with Histogram¶

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)
a, b = 2, 8
samples = stats.uniform(loc=a, scale=b-a).rvs(50_000)

fig, ax = plt.subplots(figsize=(12, 3))
ax.hist(samples, bins=60, density=True, alpha=0.7, label='Samples')
x = np.linspace(a - 1, b + 1, 300)
ax.plot(x, stats.uniform(loc=a, scale=b-a).pdf(x), 'r-', lw=2, label='PDF')
ax.spines[['top', 'right']].set_visible(False)
ax.legend()
plt.show()

Inverse Transform Sampling¶

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

np.random.seed(42)

# Generate exponential samples via inverse transform
u = np.random.uniform(0, 1, 50_000)
lam = 2.0
x_exp = -np.log(1 - u) / lam  # F_inv(u) for Exponential(lambda)

fig, ax = plt.subplots(figsize=(12, 3))
ax.hist(x_exp, bins=100, density=True, alpha=0.7, label='Inverse transform samples')
t = np.linspace(0, 4, 200)
ax.plot(t, stats.expon(scale=1/lam).pdf(t), 'r-', lw=2, label='Exponential PDF')
ax.spines[['top', 'right']].set_visible(False)
ax.legend()
plt.show()

Key Takeaways¶

The uniform distribution assigns equal probability to all values in an interval, making it the "maximally uninformative" distribution over a bounded range.
The standard uniform \(U(0,1)\) is the building block for random number generation via the inverse transform method.
The probability integral transform establishes that applying the CDF to any continuous random variable yields a uniform result.
Despite its simplicity, the uniform distribution is foundational to Monte Carlo simulation and computational statistics.