Probability

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

Measure & Probability

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

Random Variables & Measurable Functions

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