Understanding the PDF of Uniform Distribution

The uniform distribution, in its standard form, is evenly spread across the interval [0, 1]. By adjusting parameters loc and scale, this distribution can be extended to any range [loc, loc + scale]. As a subclass of rv_continuous within the SciPy library, the uniform object inherits numerous general methods, enhanced with specific functionalities tailored to this distribution.

Let’s explore the uniform distribution with practical examples.

>>> import numpy as np
>>> from scipy.stats import uniform
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

To begin, we can calculate key statistical moments—mean, variance, skew, and kurtosis—which provide insights into the shape and central tendency of the distribution.

>>> mean, var, skew, kurt = uniform.stats(moments='mvsk')

The probability density function (PDF) is crucial for understanding the likelihood of different values within the distribution. Below, we visualize the PDF using Matplotlib.

>>> x = np.linspace(uniform.ppf(0.01),
...                 uniform.ppf(0.99), 100)
>>> ax.plot(x, uniform.pdf(x),
...        'r-', lw=5, alpha=0.6, label='uniform pdf')

Alternatively, we can ‘freeze’ the distribution with fixed parameters. This creates a static distribution object, useful for repeated evaluations with the same parameters.

>>> rv = uniform()
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

To ensure the accuracy of the cumulative distribution function (CDF) and the percent point function (PPF), we can perform a check. The CDF gives the probability that a random variable from the distribution is less than or equal to a given value, while the PPF is its inverse.

>>> vals = uniform.ppf([0.001, 0.5, 0.999])
>>> np.allclose([0.001, 0.5, 0.999], uniform.cdf(vals))
True

Generating random numbers from a uniform distribution is straightforward and essential for simulations and statistical modeling.

>>> r = uniform.rvs(size=1000)

Finally, we can visually compare the generated random numbers with the theoretical distribution using a histogram, providing a visual check of our implementation.

>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

The uniform distribution, characterized by its constant probability density function across a defined interval, is a foundational concept in statistics and probability theory. Understanding its PDF, along with related functions like CDF and PPF, is crucial for both theoretical studies and practical applications. The methods available through SciPy, as outlined below, provide a comprehensive toolkit for working with uniform distributions in Python.