Sum of random variables - Check your derived distribution against a numerical calculation/histogram

Question

Consider independent random variates $X_0, X_1, . . .$ each uniformly distributed on the support $[0, 1)$

Let's say $Y = X_0 + X_1$, where $X_0$ and $X_1$ are independent uniform random variables with range $[0,1)$

I then derived the following PDF to describe the distribution of the sum $Y = X_0 + X_1$ using a convolution.

$$f(y) = \begin{cases} y & \text{for $0 < y < 1$} \\ 2-y & \text{for $1 \le y < 2$} \\ 0 & \text{otherwise.} \end{cases}$$

I want to check the derived distribution against a numerical calculation/histogram (in python)

Any help is appreciated

import numpy as np
import matplotlib.pyplot as plt

X_0 = np.random.uniform(0.0,1.0,1000)
X_1 = np.random.uniform(0.0,1.0,1000)

Y = X_0 + X_1

#Plot 3 normalised histograms
plt.hist(X_0,bins=100,density=True)
plt.hist(X_1,bins=100,density=True)
plt.hist(Y,bins=100,density=True)

#desired distribution
if 0<Y<1:
    F_Y = Y
if 1<=Y<2:
    F_Y = 2 - Y
if Y < 0:
    F_Y = 0
    
plt.plot(Y,F_Y)
plt.title('desired distibution')
plt.show()

Answer 1

Here is a working code and the result shown below, just analyze it. Note that the probability density function $p(x)$ is normalized to unity, $\int p(x) dx$=1.

import numpy as np
import matplotlib.pyplot as plt

def show_histogram(vals):
    n, bins, patches = plt.hist(x=vals, bins='auto', density=True, color='#0504aa', alpha=0.7, rwidth=0.85)
    plt.grid(axis='y', alpha=0.75)
    plt.xlabel('Value')
    plt.ylabel('Frequency')
    plt.title('Distribution of calculated values')



X_0 = np.random.uniform(0.0,1.0,10000)
X_1 = np.random.uniform(0.0,1.0,10000)

Y = X_0 + X_1


#Plot normalised histograms
show_histogram(X_0+X_1)


#desired distribution
n=100
Y = 2*np.arange(n)/(n-1)
F_Y=1.0 - np.abs(Y-1)

    
plt.plot(Y,F_Y)
plt.title('desired distibution')
plt.show()