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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

Expectation & Integral

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

Haar & Lebesgue Measure

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

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