Probability

The study of randomness and uncertainty.

• Sample Space — All possible outcomes
• Event — Subset of sample space
• Probability Axioms — Kolmogorov's axioms
• Conditional Probability — P(A|B)
• Independence — P(A∩B) = P(A)P(B)
• Bayes' Theorem — P(A|B) formula
• Random Variable — Function on sample space
• Discrete Random Variable — Countable range
• Continuous Random Variable — Uncountable range
• Probability Mass Function — P(X = x)
• Probability Density Function — f(x)
• Cumulative Distribution — F(x) = P(X ≤ x)
• Expected Value — E[X]
• Variance — E[(X-μ)²]
• Standard Deviation — sqrt(variance)
• Bernoulli Distribution — 0 or 1
• Binomial Distribution — n Bernoulli trials
• Geometric Distribution — Trials until first success
• Poisson Distribution — Rare events
• Normal Distribution — Bell curve
• Central Limit Theorem — Sum approaches normal
• Law of Large Numbers — Sample mean → μ
• Markov Inequality — P(X ≥ a) ≤ E[X]/a