Probability Theory
Understanding uncertainty and random phenomena through mathematical analysis
Foundations of Probability
Probability theory provides the mathematical framework for analyzing random phenomena, making decisions under uncertainty, and understanding statistical inference.
Core Probability Concepts
Basic Probability
- Sample Spaces and Events
- Probability Axioms
- Conditional Probability
Random Variables
- Discrete vs Continuous
- Expected Value
- Variance and Moments
Probability with Python
import numpy as np
from scipy import stats
import matplotlib.pyplot as plt
# Generate random data
n_samples = 1000
data = np.random.normal(loc=0, scale=1, size=n_samples)
# Calculate probability density
density = stats.gaussian_kde(data)
x_range = np.linspace(min(data), max(data), 200)
y = density(x_range)
# Plot probability density
plt.figure(figsize=(10, 6))
plt.hist(data, bins=30, density=True, alpha=0.7, color='blue')
plt.plot(x_range, y, 'r-', lw=2, label='KDE')
plt.title('Probability Density Estimation')
plt.xlabel('Value')
plt.ylabel('Density')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
# Calculate probabilities
prob_less_than_1 = stats.norm.cdf(1, loc=0, scale=1)
prob_greater_than_2 = 1 - stats.norm.cdf(2, loc=0, scale=1)
print(f"P(X < 1) = {prob_less_than_1:.3f}")
print(f"P(X > 2) = {prob_greater_than_2:.3f}")Advanced Topics
Stochastic Processes
- • Markov Chains
- • Poisson Processes
- • Brownian Motion
Bayesian Inference
- • Prior and Posterior
- • Conjugate Priors
- • MCMC Methods