Adding random variables

Foundations of Probability in Python

Alexander A. Ramírez M.

CEO @ Synergy Vision

The central limit theorem (CLT)

The sum of random variables tends to a normal distribution as the number of them grows to infinity.

Conditions:

The variables must have the same distribution.
The variables must be independent.

Sample generation animation for poisson distribution

Random selection from a sample plot

Sample random selection from the population

Sample mean from sample calculation and histogram

Poisson population plot

# Add the imports
from scipy.stats import poisson, describe
from matplotlib import pyplot as plt
import numpy as np

# Generate the population
population = poisson.rvs(mu=2, size=1000, random_state=20)

# Draw the histogram with labels
plt.hist(population, bins=range(9), width=0.8)
plt.show()

Poisson population histogram

Sample means plot

# Generate 350 sample means, selecting
# from population values
np.random.seed(42)

# Define list of sample means
sample_means = []
for _ in range(350):
    # Select 10 from population
    sample = np.random.choice(population, 10)
    # Calculate sample mean of sample
    sample_means.append(describe(sample).mean)

Sample means plot (Cont.)

$$ $$

# Draw histogram with labels
plt.xlabel("Sample mean values")
plt.ylabel("Frequency")
plt.title("Sample means histogram")
plt.hist(sample_means)
plt.show()

Sample mean histrogram with Bell shape

Let's add random variables

Foundations of Probability in Python