Introduction to [[Probability]] Theory

Probability theory is the mathematical framework for quantifying uncertainty. In quantitative finance, trading, and research, probability theory is the bedrock upon which models for risk, asset pricing, and algorithmic prediction are built.

Core Concepts

1. Sample Space and Events: The sample space ((\Omega)) is the set of all possible outcomes of an experiment. An event is a subset of the sample space.
2. Probability Measure: A function (P) that assigns a real number to each event, satisfying Kolmogorov's axioms:
   • (P(E) \ge 0) for any event (E).
   • (P(\Omega) = 1).
   • For mutually exclusive events (E_1, E_2, \dots), the probability of their union is the sum of their probabilities.
3. Random Variables: A random variable (X) is a function that maps outcomes in the sample space to real numbers. It allows us to work with probabilities numerically.
4. Expectation and Variance:
   • Expectation ((E)): The long-run average or mean of a random variable.
   • Variance ((Var(X))): A measure of the dispersion or spread of the random variable around its expectation.
5. Distributions: Describes how the probabilities are distributed over the possible values of the random variable. Key distributions include the Normal (Gaussian), Binomial, Poisson, and Lognormal distributions.
6. Conditional [[Probability and Bayes' Theorem]]: The probability of an event given that another event has occurred. Bayes' Theorem provides a way to update probabilities based on new evidence.

Understanding these foundational concepts is crucial for advancing to advanced topics like statistics, machine learning, and stochastic calculus.