# Basics of Probability

Some notes on foundational ideas and definitions in modern probability theory.

• ### Sample Spaces, Events & $\sigma$-Algebras

The set of possible outcomes of a random experiment is called the sample space. Subsets of the sample space are called events. We will eventually axiomatize probability measures, but for the moment we view probability as a number $\mathbb P(E) \in [0,1]$ we associate to certain events which represents the likelihood that an outcome of… Read More »Sample Spaces, Events & $\sigma$-Algebras

• ### Conditional Expectation

Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space, and suppose $\mathcal F \subseteq \mathcal H$ is a sub-$\sigma$-algebra of $\mathcal H$. Then, if $X$ is a random variable measurable with respect to $\mathcal F$, then it is measurable with respect to $\mathcal H$, but the converse is not true. The central question that… Read More »Conditional Expectation

• ### Random Processes & Random Fields

Let $\mathbb T$ be an index set, usually a subset of $\mathbb R^N$ for some $N$. Then a collection of random variables $\{X_t\}_{t \in \mathbb T}$ on the same probability space $(\Omega, \mathcal H, \mathbb P)$ is a random process. From another perspective, for each $\omega \in \Omega$ we have $t \rightarrow X_t(\omega)$ is a… Read More »Random Processes & Random Fields