Probability and Random Numbers

Duration: 8 min

Try it in Google Colab:

Random Number Generation

import numpy as np

# Set seed for reproducibility
np.random.seed(42)

# Uniform distribution [0, 1)
uniform = np.random.rand(5)

# Normal distribution (mean=0, std=1)
normal = np.random.randn(5)

# Random integers
integers = np.random.randint(0, 10, 5)

# Random choice from array
choices = np.random.choice([1, 2, 3, 4, 5], 5)

Probability Distributions

# Uniform distribution
uniform_samples = np.random.uniform(0, 10, 1000)

# Normal distribution
normal_samples = np.random.normal(loc=0, scale=1, size=1000)

# Binomial distribution
binomial_samples = np.random.binomial(n=10, p=0.5, size=1000)

# Poisson distribution
poisson_samples = np.random.poisson(lam=3, size=1000)

Computing Probabilities

# Simulate coin flips
flips = np.random.randint(0, 2, 10000)

# Probability of heads
p_heads = np.sum(flips) / len(flips)
print(f"P(Heads) ≈ {p_heads}")  # ≈ 0.5

# Simulate dice rolls
rolls = np.random.randint(1, 7, 10000)

# Probability of rolling 6
p_six = np.sum(rolls == 6) / len(rolls)
print(f"P(6) ≈ {p_six}")  # ≈ 0.167

Expected Value and Variance

# Generate samples
samples = np.random.normal(loc=5, scale=2, size=10000)

# Expected value (mean)
expected_value = np.mean(samples)
print(f"E[X] ≈ {expected_value}")  # ≈ 5

# Variance
variance = np.var(samples)
print(f"Var(X) ≈ {variance}")  # ≈ 4

# Standard deviation
std_dev = np.std(samples)
print(f"σ ≈ {std_dev}")  # ≈ 2

Sampling and Estimation

# Estimate π using Monte Carlo
n_samples = 100000

# Generate random points in [0,1] x [0,1]
x = np.random.uniform(0, 1, n_samples)
y = np.random.uniform(0, 1, n_samples)

# Distance from origin
distances = np.sqrt(x**2 + y**2)

# Points inside unit circle
inside_circle = np.sum(distances <= 1)

# Estimate π
pi_estimate = 4 * inside_circle / n_samples
print(f"π ≈ {pi_estimate}")  # ≈ 3.14159

Correlation and Covariance

# Generate correlated variables
x = np.random.randn(1000)
y = 2 * x + np.random.randn(1000)

# Covariance
cov = np.cov(x, y)
print("Covariance matrix:\n", cov)

# Correlation
corr = np.corrcoef(x, y)
print("Correlation matrix:\n", corr)