# Summing Uniform Distributions¶

The distribution of the sum of N uniformly distributed variables came up in a recent RdTools PR. The approach taken there was to just sample each distribution and sum the samples, but I wondered how it could be done analytically. This notebook is inspired by this StackExchange post and extends the derivation from just summing two uniform distributions on $$[0,1]$$ to two distributions with arbitrary bounds.

Given two independent random variables $$A$$ and $$B$$ with probability densities $$f_a(x)$$ and $$f_b(x)$$, the probability density $$f_c(x)$$ of their sum $$A + B = C$$ is given by the convolution $$f_a * f_b$$:

$f_c(x) = \int_{-\infty}^{\infty} f_a(x') f_b(x - x') \mathrm{d}x'$

For the case where $$A$$ and $$B$$ are uniformly distributed, we have:

$\begin{split} f_a(x) = \begin{cases} \frac{1}{a_2 - a_1} & a_1 \le x \le a_2 \\ 0 & \mathrm{otherwise} \end{cases} \end{split}$
$\begin{split} f_b(x) = \begin{cases} \frac{1}{b_2 - b_1} & b_1 \le x \le b_2 \\ 0 & \mathrm{otherwise} \end{cases} \end{split}$

Using these definitions, let’s visualize in the $$X'-X$$ plane where the integrand is nonzero:

import matplotlib.pyplot as plt
import numpy as np

a1 = 0.2
a2 = 1.0
b1 = 1.0
b2 = 2.5

Xlim = [1, 4]
Xplim = [0, 1.2]
Xp, X = np.meshgrid(np.linspace(*Xplim, 1000), np.linspace(*Xlim, 1000))
nonzero = (a1 <= Xp) & (Xp <= a2) & (b1 <= X - Xp) & (X - Xp <= b2)

plt.imshow(nonzero, extent=Xplim + Xlim, aspect='auto', origin='lower')
plt.xlabel("x'")
plt.ylabel("x")

plt.axhline(b1 + a1, c='cyan')
plt.axhline(b1 + a2, c='cyan')
plt.axhline(b2 + a1, c='cyan')
plt.axhline(b2 + a2, c='cyan');


The cross-section of the parallelogram is the range of $$x'$$ where the integrand is nonzero, so the lower and upper bounds of each cross-section define the limits of integration as a function of $$x$$. Notice that it can be divided into three domains with nonzero probability, drawn here separated by the cyan lines. Going up from the bottom, they are defined by:

Domain

Domain bounds

Integration bounds

1

$$b_1 + a_1 \le x \le b_1 + a_2$$

$$a_1 \le x' \le x - b_1$$

2

$$b_1 + a_2 \le x \le b_2 + a_1$$

$$a_1 \le x' \le a_2$$

3

$$b_2 + a_1 \le x \le b_2 + a_2$$

$$x - b_2 \le x' \le a_2$$

Finally, note that with the restricted bounds of integration $$x_1$$ and $$x_2$$, the integrand becomes the constant $$(a_2-a_1)^{-1}(b_2-b_1)^{-1}$$ and so the integral simplifies:

\begin{split} \begin{align} f_c(x) &= \frac{1}{(a_2-a_1)(b_2-b_1)} \int_{x'_1}^{x'_2} \mathrm{d}x' \\ &= \frac{x'_2 - x'_1}{(a_2-a_1)(b_2-b_1)} \end{align} \end{split}

where $$x'_1$$ and $$x'_2$$ are given by the integration bounds in the above table as a function of $$x$$.

Note that we have assumed $$b_1 + a_2 \le b_2 + a_1$$. If that is not the case, swap the $$a$$s and $$b$$s.

# Numerical approach
N = 10000000
A = np.random.uniform(a1, a2, N)
bd = 1/(b2-b1)
B = np.random.uniform(b1, b2, N)
C = A + B

plt.hist(C, bins=100, density=True, label='numerical PDF')

# Analytical approach
scale = 1/(a2-a1)/(b2-b1)

def f_c(x, a1, a2, b1, b2):
if a1 + b2 < b1 + a2:
a1, a2, b1, b2 = b1, b2, a1, a2

y = np.zeros_like(x)
region1 = (b1 + a1 <= x) & (x <= b1 + a2)
region2 = (b1 + a2 <= x) & (x <= b2 + a1)
region3 = (b2 + a1 <= x) & (x <= b2 + a2)
y[region1] = scale * (x[region1] - b1 - a1)
y[region2] = scale * (a2 - a1)
y[region3] = scale * (a2 - (x[region3] - b2))
return y

x = np.linspace(a1+b1, a2+b2, 1000)
y = f_c(x, a1, a2, b1, b2)
plt.plot(x, y, label='analytical PDF')
plt.xlabel('x')
plt.legend();


Going back to the original problem of summing not just two random variables but $$N$$ of them, it seems reasonable to expect that increasing $$N$$ will increase the order of the polynomials involved, and visualizing it for $$N=3$$ does show some higher-order behavior:

A = np.random.uniform(a1, a2, N)
bd = 1/(b2-b1)
B = np.random.uniform(b1, b2, N)
C = np.random.uniform(1.6, 2.0, N)
D = A + B + C

plt.hist(D, bins=100, density=True, label='numerical PDF')
plt.legend();