Skip to content


Joint Distributions

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

Independence & Conditioning

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

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… Read More »Conditional Expectation

Point Processes

Let $(E, \mathcal E)$ be a measurable space, and let $(E^N, \mathcal E^{\otimes N})$ be the product space equipped with the product $\sigma$-algebra. We will view $\mathbf x = (x_1, \ldots, x_N)$ as the position… Read More »Point Processes