Distribution function of $X_1 + X_2$ given probability distributions of $X_1$ and $X_2$

Question

Suppose I have a probability distribution for random variable $X_1$ given by:

$$ P(x) = \begin{cases} 1/a, & \text{if} \;\; 0 \leq x \leq a\\ 0, & \text{otherwise} \end{cases} $$

and $X_2$ with probability distribution:

$$ P(x) = \begin{cases} 1/b, & \text{if} \;\; 0 \leq x \leq b\\ 0, & \text{otherwise} \end{cases} $$

What is the probability distribution function of $X_1 + X_2$.

Answer 1

These are both uniform random variables if they independent we can write it like this. First $X_{1} \sim U(0,a)$ $X_{2} \sim U(0,b)$ . Also note the proof assumes that $ a \leq b$ The pdf of a uniform random variable $X \sim U(a,b)$ is given by

$$ f_{X}(x) =\begin{align}\begin{cases} \frac{1}{b-a} & \textrm{ for } x \in [a,b] \\ \\ 0 & \textrm{ for everywhere else } \end{cases} \end{align} \tag{1}$$

then the pdf of $X_{1}$ is given by

$$f_{X_{1}}(x_{1}) =\begin{align}\begin{cases} \frac{1}{a} & \textrm{ for } x \in [0,a] \\ \\ 0 & \textrm{ for everywhere else } \end{cases} \end{align}\tag{2}$$

so the pdf of $X_{2}$ is $$f_{X_{2}}(x_{2}) =\begin{align}\begin{cases} \frac{1}{b} & \textrm{ for } x \in [0,b] \\ \\ 0 & \textrm{ for everywhere else } \end{cases} \end{align} \tag{3}$$

Let the density $S = X_{1} + X_{2}$ be given by

$$ f_{S}(s) = \int_{\mathbb{R}} f_{X_{1}}(s-x_{2})f_{X_{2}}(x_{2}) dx_{2} \tag{4}$$

$$f_{X_{1}}(s-x_{2}) =\begin{align}\begin{cases} \frac{1}{a} & 0 \leq s-x_{2} \leq a \\ \\ 0 & \textrm{ for everywhere else } \end{cases} \end{align} \tag{5}$$ $$f_{X_{2}}(x_{2}) =\begin{align}\begin{cases} \frac{1}{b} & 0 \leq x_{2} \leq b \\ \\ 0 & \textrm{ for everywhere else } \end{cases} \end{align} \tag{6}$$

when $ 0 \leq x_{2} \leq a$

$$ f_{S}(s) = \int_{0}^{s} \frac{1}{ab} dx_{2} = \frac{x_{2}}{ab}\Big|_{0}^{s} = \frac{s}{ab} \tag{7}$$ when $ a \leq x_{2} < b $ $$ f_{S}(s) =\int_{s-a}^{s} \frac{1}{ab} dx_{2} = \frac{1}{ab}\Big|_{s-a}^{s} = \frac{s}{ab} - \frac{s-a}{ab}= \frac{1}{ab}(s-(s-a)) = \frac{a}{ab} =\frac{1}{b} \tag{8} $$

when $ b \leq x_{2} \leq a+b $

$$ f_{S}(s) = \int_{s-a}^{b} \frac{1}{ab} dx_{2} =\frac{1}{ab}\Big|_{s-a}^{b} = \frac{b}{ab} - \frac{s-a}{ab} = \frac{1}{a} - \frac{s-a}{ab} = \frac{1}{a} - \frac{s}{ab}+\frac{1}{b} \tag{9}$$

$$f_{S}(s) =\begin{align}\begin{cases} \frac{1}{ab}s & 0 \leq s < a \\ \\ \frac{1}{b} & a \leq s < b \\ \frac{1}{a} -\frac{s}{ab}+\frac{1}{b} &b\leq s < a+b \\ 0 & \textrm{ otherwise} \end{cases} \end{align} \tag{10}$$

In order to visualize this here is some code

import numpy as np
import matplotlib.pyplot as plt


def sum_uni(a,b,n):

    z1 = np.linspace(0,a,n)
    z2 = np.ones(n)   
    z3 = np.linspace(b,a+b,n) 

    f1 = (1/(a*b))*z1
    f2 = (1/b)*z2
    f3 = (1/a)*z2-(1/(a*b))*z3 +(1/b)*z2
    g1 = np.linspace(0,a,n)
    g2 = np.linspace(a,b,n)
    g3 = np.linspace(b,a+b,n)
    my_plot = np.concatenate((g1,g2,g3),axis=0)
    my_distribution = np.concatenate((f1,f2,f3),axis=0)

    return my_plot, my_distribution

my_plot, my_dist = sum_uni(2,3,100)

plt.plot(my_plot,my_dist)

Answer 2

Here is a figure based on a simulation of a million realizations of $X = U_1 + U_2,$ where $U_1 \sim \mathsf{Unif}(0, a),\, U_2 \sim \mathsf{Unif}(0, b),$ independently, with $a = 1,\, b= 3.$

set.seed(9118);  m = 10^6;  a=1;  b=3
u1 = runif(m, 0, a);  u2 = runif(m, 0, b)
x = u1 + u2
hist(x, br=50, prob=T, col="skyblue2")
 curve(x/(a*b), 0, a, col="red", lwd=2, add=T)
 lines(c(a, b), c(1/b, 1/b), col="red", lwd=2)
 curve(1/a + 1/b - x/(a*b), b, a+b, col="red", lwd=2, add=T)