Uniform Distributions¶

NumPy provides functions for generating uniformly distributed random numbers over continuous intervals.

np.random.rand¶

Generates samples uniformly distributed over \([0, 1)\).

1. Basic Usage¶

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

def main():
    np.random.seed(0)

    n_samples = 10_000
    data = np.random.rand(n_samples)

    fig, ax = plt.subplots()

    _, bins_, _ = ax.hist(data, bins=100, density=True)

    low_ = data.min()
    high_ = data.max()
    pdf_at_bins_ = stats.uniform(loc=low_, scale=high_ - low_).pdf(bins_)
    ax.plot(bins_, pdf_at_bins_, '--r', linewidth=5)

    plt.show()

if __name__ == "__main__":
    main()

2. Shape Arguments¶

Pass dimensions as separate arguments: np.random.rand(3, 2) for a 3×2 array.

np.random.uniform¶

Generates samples uniformly distributed over a specified interval.

1. Custom Interval¶

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

def main():
    np.random.seed(0)

    low = -1
    high = 1
    n_samples = 10_000

    data = np.random.uniform(low=low, high=high, size=(n_samples,))

    fig, ax = plt.subplots()

    _, bins_, _ = ax.hist(data, bins=100, density=True)

    low_ = data.min()
    high_ = data.max()
    pdf_at_bins_ = stats.uniform(loc=low_, scale=high_ - low_).pdf(bins_)
    ax.plot(bins_, pdf_at_bins_, '--r', linewidth=5)

    plt.show()

if __name__ == "__main__":
    main()

2. Half-Open Interval¶

Samples are drawn from \([\text{low}, \text{high})\), excluding the upper bound.

Scaling Relation¶

Any uniform distribution can be derived from \(U(0, 1)\).

1. Linear Transform¶

\(X \sim U(a, b)\) is equivalent to \(X = a + (b - a) \cdot U\) where \(U \sim U(0, 1)\).

2. Practical Choice¶

Use rand for \([0, 1)\) and uniform for custom intervals.

PDF Shape¶

The uniform distribution has constant probability density.

1. Flat Histogram¶

A properly normalized histogram of uniform samples appears flat.

2. Theoretical PDF¶

\[f(x) = \frac{1}{b - a} \quad \text{for } x \in [a, b)\]

Common Applications¶

Uniform random numbers have many practical uses.

1. Random Selection¶

Uniformly sample indices or elements from arrays.

2. Monte Carlo¶

Uniform samples over \([0, 1)\) are the basis for many simulation methods.

3. Initialization¶

Neural network weights are often initialized from uniform distributions.