Probability

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… Read More »Sample Spaces, Events & $\sigma$-Algebras

We begin with the sample space $\Omega$ of a random experiment, and a $\sigma$-algebra $\mathcal H$ on $\Omega$ consisting of subsets of $\Omega$ to which we want to assign probabilities. Definition: A probability measure on… Read More »Measure & Probability

A measurable space consists of a set and a $\sigma$-algebra on that set. If $(E, \mathcal E)$ and $(F, \mathcal F)$ are measurable spaces, then we say $f: E \rightarrow F$ is measurable if for… Read More »Random Variables & Measurable Functions

Given a probability space $(\Omega, \mathcal H, \mathbb P)$ and positive simple random variable $X \in \mathcal H_+$, $$X = \sum_n a_n \boldsymbol 1_{A_n},$$ the expectation of $X$ is given by $$\mathbb E[X] = \sum_n… Read More »Expectation & Integral

Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space and let $(E, \mathcal E, \mu)$ be a measure space. Set $A \in \mathcal H$ and $B \in \mathcal E$ are called null sets if… Read More »Almost Surely, Almost Never & Almost Everywhere

Our expectations and integrals are examples of Lebesgue integrals. How do these relate to the Riemann integrals we learned about in calculus? The group of real numbers under addition $(\mathbb R, +)$ is an example… Read More »Haar & Lebesgue Measure

Suppose $(\Omega, \mathcal H, \mathbb P)$ is a probability space and let $X$ be a random variable. Then, we may define a measure on $(\mathbb R, \mathcal B(\mathbb R))$ by setting $$\mu_X(A) = \mathbb P\{X… Read More »Distributions & Densities

Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space and consider a random vector $\mathbf X: \Omega \rightarrow \mathbb R^N$. By calling $\mathbf X$ we assume that each coordinate $\mathbf X_n : \Omega \rightarrow… Read More »Joint Distributions

Here we introduce a purely probabilistic concept, independence. Let $(\Omega, \mathcal H, \mathbb P)$ be a probability space. Definition: Given $A, B \in \mathcal H$ with $\mathbb P(B) \neq 0$, we define the conditional probability… Read More »Independence & Conditioning

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… Read More »Conditional Expectation