Uniform Distribution¶

The uniform distribution is the simplest continuous probability distribution, assigning equal probability density to every point in an interval \([a, b]\). Despite its simplicity, it plays a foundational role in probability theory and computational statistics. The probability integral transform states that applying the CDF of any continuous distribution to a random variable from that distribution yields a uniform random variable, making the uniform distribution the starting point for all random variate generation methods.

Mathematical Definition¶

A random variable \(X\) follows a continuous uniform distribution on \([a, b]\), written \(X \sim U(a, b)\), if its probability density function is:

The cumulative distribution function is:

with \(F(x) = 0\) for \(x < a\) and \(F(x) = 1\) for \(x > b\).

Parametrization in scipy.stats¶

An important detail: scipy.stats.uniform uses loc and scale rather than \(a\) and \(b\) directly. The support is \([\text{loc},\; \text{loc} + \text{scale}]\), so to represent \(U(a, b)\) you pass loc=a and scale=b-a:

```python import scipy.stats as stats import numpy as np import matplotlib.pyplot as plt

a, b = 2.0, 7.0 dist = stats.uniform(loc=a, scale=b - a)

print(f"Mean: {dist.mean():.4f}") # (a+b)/2 = 4.5 print(f"Variance: {dist.var():.4f}") # (b-a)²/12

x = np.linspace(0, 9, 200) y_pdf = dist.pdf(x) y_cdf = dist.cdf(x)

plt.plot(x, y_pdf, label='PDF') plt.plot(x, y_cdf, label='CDF') plt.legend() plt.title(f'Uniform Distribution U({a}, {b})') plt.xlabel('x') plt.ylabel('Density / Probability') plt.show() ```

The PDF is a flat rectangle of height \(1/(b-a)\) between \(a\) and \(b\), and the CDF is a straight line rising from 0 to 1 over that interval.

Standard Uniform U(0, 1)¶

The standard uniform distribution \(U(0, 1)\) is the default when no parameters are specified:

```python import scipy.stats as stats

u = stats.uniform() # U(0, 1) by default print(f"Mean: {u.mean():.4f}") # 0.5 print(f"Variance: {u.var():.4f}") # 1/12 ≈ 0.0833 print(f"P(X ≤ 0.3): {u.cdf(0.3):.4f}") # 0.3 ```

Key Properties¶

• Mean: \(E[X] = \dfrac{a + b}{2}\)
• Variance: \(\text{Var}(X) = \dfrac{(b - a)^2}{12}\)
• Symmetry: The distribution is symmetric about the midpoint \((a + b)/2\)
• Maximum entropy: Among all continuous distributions on \([a, b]\), the uniform distribution has the maximum entropy (it encodes the least information)

Parameters in scipy.stats¶

Probability Integral Transform¶

The probability integral transform is a fundamental result connecting any continuous distribution to the uniform. If \(X\) is a continuous random variable with CDF \(F\), then:

Conversely, if \(U \sim U(0, 1)\) and \(F\) is any continuous CDF, then \(X = F^{-1}(U)\) has CDF \(F\). This is called inverse transform sampling and is the basis for generating random variates from any distribution:

```python import scipy.stats as stats import numpy as np import matplotlib.pyplot as plt

Generate exponential(λ=2) samples via inverse transform¶

u = stats.uniform.rvs(size=5000, random_state=42) lam = 2.0 x_inv = -np.log(1 - u) / lam # F⁻¹(u) for exponential

Compare with direct sampling¶

x_direct = stats.expon(scale=1/lam).rvs(size=5000, random_state=43)

plt.hist(x_inv, bins=40, density=True, alpha=0.5, label='Inverse transform') plt.hist(x_direct, bins=40, density=True, alpha=0.5, label='Direct sampling') plt.legend() plt.title('Exponential Samples via Inverse Transform vs Direct') plt.xlabel('x') plt.ylabel('Density') plt.show() ```

Financial Applications¶

In quantitative finance, the uniform distribution appears in Monte Carlo simulation (all pseudorandom number generators start by producing uniform variates). Copula models use the probability integral transform to convert marginal distributions to uniform margins before applying a dependence structure. Quasi-random sequences (Sobol, Halton) generate low-discrepancy uniform samples for variance reduction in pricing.

Summary¶

The uniform distribution assigns equal density across an interval and serves as the foundation for random variate generation through the probability integral transform. In scipy.stats, use stats.uniform(loc=a, scale=b-a) to represent \(U(a, b)\), noting the loc-scale parametrization rather than the endpoint parametrization.

Exercises¶

Exercise 1. Create a uniform distribution on \([3, 8]\) using scipy.stats.uniform. Compute \(P(4 \le X \le 6)\) using the CDF and verify it equals \((6 - 4) / (8 - 3)\).

from scipy import stats

rv = stats.uniform(loc=3, scale=5)
prob = rv.cdf(6) - rv.cdf(4)
expected = (6 - 4) / (8 - 3)
print(f"P(4 <= X <= 6) = {prob:.4f}, expected = {expected:.4f}")

Exercise 2. Generate 10,000 uniform samples on \([0, 1]\) and apply the CDF of an exponential distribution with \(\lambda = 2\) (the probability integral transform in reverse) to produce exponential samples. Verify by comparing the sample mean to \(1/\lambda\).

import numpy as np
from scipy import stats

np.random.seed(42)
u = np.random.uniform(0, 1, size=10000)
exp_samples = stats.expon.ppf(u, scale=0.5)
print(f"Sample mean: {np.mean(exp_samples):.4f} (expected 0.5)")

Exercise 3. Compute the entropy of uniform distributions on \([0, 1]\), \([0, 10]\), and \([0, 100]\). Verify that the entropy equals \(\ln(b - a)\).

import numpy as np
from scipy import stats

for width in [1, 10, 100]:
    rv = stats.uniform(loc=0, scale=width)
    print(f"U[0,{width:3d}]: entropy={rv.entropy():.4f}, ln({width})={np.log(width):.4f}")