1 Packages
2 Continuous Uniform Distributions
- 2.1 Probability Density Function
- 2.2 Cumulative Distribution Function
3 Discrete Uniform Distributions
- 3.1 Probability Mass Function
- 3.2 Cumulative Distribution Function

If someone ever wants to generate a ‘random number’ it usually means that they want a random number from a uniform distribution: they want the chance of getting any one number to be the same as the chance of getting any other number. The chances of particular numbers appearing is uniform - hence the name. They will often also want that random number to be an integer, which implies that they want it to come from a discrete uniform distribution. If they want that random number to be any decimal, it implies they want it to come from a continuous uniform distribution.

You might be aware that, in general, computers can’t actually produce truly random numbers and can only produce pseudorandom numbers. For the purposes of this page this fact makes no difference.

1 Packages

The code on this page uses the numpy, matplotlib and scipy packages. These can be installed from the terminal with:

$ python3.11 -m pip install numpy
$ python3.11 -m pip install matplotlib
$ python3.11 -m pip install scipy

where python3.11 corresponds to the version of Python you have installed and are using.

Import these packages into Python with:

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

2 Continuous Uniform Distributions

A number sampled from a continuous uniform distribution that runs from \(A\) to \(B\) can have any value between those two endpoints with no value being more or less likely than any other possible value.

If the interval is closed then the sampled number can have any value between \(A\) and \(B\) but not \(A\) or \(B\) exactly. This is denoted by square brackets: \([A, B]\).
If the interval is open then the sampled number can have any value between \(A\) and \(B\) including \(A\) or \(B\) exactly. This is denoted by round brackets: \((A, B)\).
Half-open intervals - \((A, B]\) or \([A, B)\) - are a mixture: the random number could be equal to the endpoint on the open side but not the endpoint on the closed side.

In Python, the uniform() function from SciPy samples from a closed interval by default while the uniform() function from NumPy samples from a half-open one - \([A, B)\) - by default. SciPy also gives us access to the probability density function (PDF), cumulative distribution function (CDF) and percent point function (PPF) while NumPy simply lets us draw random numbers.

2.1 Probability Density Function

The PDF from SciPy:

# We want to display the PDF on the plot from -1.5 to 11.5
x = np.linspace(-1.5, 11.5, 1000)
# Probability density function
# (scaled up so as to run from 0 to 10)
pdf = st.uniform.pdf(x, loc=0, scale=10)

Random numbers from NumPy:

# Set the seed so that we get the same random numbers each time this code runs
np.random.seed(20230809)

# Pick 300 random numbers between 0 and 10
n = 300
random_numbers = np.random.uniform(0, 10, n)

Here’s what they look like on a plot:

# Create plot
ax = plt.axes()
title = """Probability Density Function (PDF) of
a Continuous Uniform Distribution"""
ax.set_title(title)
ax.set_ylabel('Relative Likelihood')
ax.set_xlabel('Potential Value of a Random Variable')
# Plot the random numbers
label = '300 Random Numbers'
ax.hist(random_numbers, density=True, label=label, color='gray', alpha=0.5)
# Plot the PDF
ax.plot(x, pdf, c='k', lw=2, label='PDF')
# Move the left axis
ax.spines['left'].set_position(('data', 0))
# Remove the upper and right axes
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
# Adjust axes
plt.ylim([0, 0.14])
# Finish
ax.legend(frameon=False)
plt.show()

2.2 Cumulative Distribution Function

The CDF from SciPy:

# We want to display the CDF on the plot from -1.5 to 11.5
x = np.linspace(-1.5, 11.5, 1000)
# Cumulative distribution function
# (scaled up so as to run from 0 to 10)
cdf = st.uniform.cdf(x, loc=0, scale=10)

Area under the plot of the random numbers from NumPy:

# Cumulative sum of random numbers
random_numbers.sort()
auc = [np.trapz([0.1] * i, random_numbers[:i]) for i in range(300)]

In a plot:

# Create plot
ax = plt.axes()
title = """Cumulative Distribution Function (CDF) of
a Continuous Uniform Distribution"""
ax.set_title(title)
ax.set_ylabel('Cumulative Relative Likelihood')
ax.set_xlabel('Potential Value of a Random Variable')
# Plot the CDF
ax.plot(x, cdf, c='k', lw=2, label='CDF')
# Plot the cumulative sum of the random numbers
label = '300 Random Numbers'
ax.hist(
    random_numbers, bins=40, density=True, cumulative=True,
    histtype='step', label=label
)
# Move the left axis
ax.spines['left'].set_position(('data', 0))
# Remove the upper and right axes
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
# Adjust axes
plt.ylim([0, 1.2])
# Finish
ax.legend(frameon=False)
plt.show()

See also the Wikipedia page, SciPy documentation and NumPy documentation relevant to the continuous uniform distribution.

3 Discrete Uniform Distributions

This concept is exactly the same as continuous distributions except now there are only a finite number of possibilities that the values of the random numbers can have. The classic example is a die roll: there are only 6 values that a roll can result in and all 6 are equally likely. The fact that the probability of getting a particular exact value is not zero (as is the case with continuous distributions) means that the probability density is not relevant, the probability mass is. In other words, we work with exact probabilities, not relative likelihoods.

3.1 Probability Mass Function

The PMF from SciPy:

# We want to display the PMF on the plot from 0 to 7
x = np.linspace(0, 7, 701)
# Probability mass function
# (running from 1 to 6)
pmf = st.randint.pmf(x, low=1, high=7)