probability

# Probability ## What I Do I provide comprehensive probability theory tools including probability distributions, conditional probability, Bayes' theorem, random variables, expectation calculations, and stochastic process simulations for uncertainty quantification. ## When to Use Me - Modeling uncertain phenomena - Bayesian inference and updating beliefs - Risk assessment and decision making - Monte Carlo simulations - Queueing theory applications - Machine learning probabilistic models ## Core Concepts - **Probability Axioms**: Kolmogorov's axioms, sample spaces - **Random Variables**: Discrete and continuous distributions - **Conditional Probability**: P(A|B) = P(A∩B)/P(B) - **Bayes' Theorem**: P(H|E) = P(E|H) * P(H) / P(E) - **Expectation & Variance**: E[X], Var(X), moment generating functions - **Convergence**: Law of large numbers, Central Limit Theorem - **Stochastic Processes**: Markov chains, Poisson processes - **Information Theory**: Entropy, KL divergence ## Code Examples ### Basic Probability Calculations ```python from scipy import stats import numpy as np P_A = 0.3 P_B = 0.4 P_A_and_B = 0.12 P_A_or_B = P_A + P_B - P_A_and_B P_A_given_B = P_A_and_B / P_B print(f"P(A or B): {P_A_or_B}") print(f"P(A|B): {P_A_given_B}") ``` ### Bayes' Theorem Application ```python prior_disease = 0.01 sensitivity = 0.95 specificity = 0.90 likelihood_positive = sensitivity likelihood_negative = 1 - specificity evidence = likelihood_positive * prior_disease + (1 - specificity) * (1 - prior_disease) posterior = (likelihood_positive * prior_disease) / evidence print(f"P(Disease|Positive): {posterior:.4f}") ``` ### Distribution Sampling ```python np.random.seed(42) normal_samples = np.random.normal(0, 1, 1000) exponential_samples = np.random.exponential(1, 1000) poisson_samples = np.random.poisson(5, 1000) binomial_samples = np.random.binomial(20, 0.5, 1000) print(f"Normal - Mean: {np.mean(normal_samples):.3f}, Std: {np.std(normal_samples):.3f}") print(f"Exponential - Mean: {np.mean(exponential_samples):.3f}") print(f"Poisson - Mean: {np.mean(poisson_samples):.3f}") ``` ### Expectation and Variance ```python from scipy.stats import norm, expon X = norm(0, 1) expected_value = X.mean() variance = X.var() print(f"E[X] for N(0,1): {expected_value}") print(f"Var(X) for N(0,1): {variance}") custom_dist = expon(scale=2) print(f"E[X] for Exp(2): {custom_dist.mean()}") ``` ### Monte Carlo Simulation ```python np.random.seed(42) n_simulations = 100000 def estimate_pi(n): x = np.random.uniform(-1, 1, n) y = np.random.uniform(-1, 1, n) inside = (x**2 + y**2) <= 1 return 4 * inside.sum() / n pi_estimate = estimate_pi(n_simulations) print(f"Pi estimate: {pi_estimate:.5f}") print(f"Error: {abs(pi_estimate - np.pi):.5f}") ``` ## Best Practices 1. **Check Conditions**: Verify assumptions before applying theorems 2. **Numerical Stability**: Use log-probabilities for small probabilities 3. **Conjugate Priors**: Use for computational efficiency in Bayesian analysis 4. **Law of Large Numbers**: Use adequate sample sizes 5. **Sampler Validation**: Check MCMC convergence diagnostics ## Common Patterns ```python # Markov chain simulation def markov_chain_transition(transition_matrix, initial_state, n_steps): n_states = transition_matrix.shape[0] current = initial_state states = [current] for _ in range(n_steps): current = np.random.choice(n_states, p=transition_matrix[current]) states.append(current) return states # Conditional probability calculation def conditional_probability(joint_prob, marginal_prob): return joint_prob / marginal_prob # KL divergence def kl_divergence(p, q): return np.sum(p * np.log(p / q)) ``` ## Core Competencies 1. Probability axioms and rules 2. Bayes' theorem and Bayesian inference 3. Probability distributions and sampling 4. Expectation and moment calculations 5. Monte Carlo methods and simulation