Random Variable

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Random Variable

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Definition

ProbabilityStochasticProcessesInfoTheory

Let $(\Omega,\mathcal{F},P)$ be a Probability Space. A random variable is a measurable function $\begin{align*} &X:(\Omega,\mathcal{F})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))\\ &\omega\in\Omega\mapsto X(\omega)\in\mathbb{R} \end{align*}$ i.e. for any $t\in\mathbb{R}$ , the set $\{\omega\in \Omega: X(\omega)\le t\}\in\mathcal{F}$ (i.e. is an event).

1. If $A\in\mathcal{F}$ , then for $X(\omega)=\mathbb{1}_{A}(\omega)$ , we have that $X$ is an rv 2. If $X,Y$ are rvs and $c$ is some constant, then $X+c,cX,XY,X+Y$ are rvs. 3. If $z_{1},z_{2}\dots$ are rvs, then $\inf_{n}z_{n},\sup_{n}z_{n},\limsup_{n\to \infty}z_{n},\liminf_{n\to \infty}z_{n},\lim_{ n \to \infty } z_{n}(\omega)=z(\omega) \,(\forall\omega \in\Omega)$ are all $rvs$ .

For a rv $X$ and a Borel function $f:\mathbb{R}\to \mathbb{R}$ we have that $f(X):\Omega\to \mathbb{R}$ is a rv.

Linked from

Continuous-time Gaussian process motion planning via probabilistic inference

Change of Variable Formula

Beppo Levi Theorem

Fubini-Tonelli

Weak convergence

Admissible Policy

Controlled Markov Chain

Blackwell's Irrelevant Information Theorem

Kalman Filter

Information Signal

Static Quadratic Team

Stationary Radner Krainak Theorem

Bayesian Statistics

Likelihood

Log-likelihood

Redundancy

Bit Allocation Problem

Continuous Memoryless Source

Differential Divergence

Differential Entropy

Equivalent Properties in Discrete & Continuous IT

Estimation Error and differential entropy

Uniform Quantization of Real-Valued Source

Discrete Memoryless Source

Entropy

Joint Entropy

Mutual Information

Renyi Entropy

Source Entropy

Source

Data Processing Inequality

Closed-loop Predictor Coefficients

Difference Quantization

Linear Prediction

Wide Sense Stationary Process

Almost Sure Convergence

Convergence in Distribution

Convergence in Expectation

In Probability Convergence

Pointwise Convergence

Central Limit Theorem

Existence of Sequences of Independent rvs

Law of Large Numbers

Portmanteau's Theorem

Scheffé's Theorem

Skohorod's Theorem

Conditional Expectation

Conditional Variance

Covariance

Expectation

Moment

Orthogonal

Variance

Cauchy-Schwartz Inequality

Expectation of a Function of a Random Variable

Markov's Inequality

Distribution

Exchangeable

Independent

Random Variable

Random Vector

iid

σ(X)

De Finetti's Theorem

Kolmogorov 0-1 Law

(Cumulative) Distribution Function

Conditional Probability Density Function

Conditional Probability Mass Function

Joint Probability Density Function

Joint Probability Mass Function

Marginal Probability Density Function

Marginal Probability Mass Function

Probability Density Function

Probability Mass Function

Summary of MATH 895

Binomial Random Variable

Exponential Random Variable

Gamma Random Variable

Gaussian Random Variable

Geometric Random Variable

Negative Binomial RV

Poisson Random Variable

Standard Normal Random Variable

Memoryless Property 1

Uniform Random Variable

(Λ) Set of Predictable Locally Integrable Processes

Replicating Portfolio

Conditional Independence

Monte Carlo Method

Doob's Upcrossing Inequality

Martingale Convergence Theorem

Dirac Distribution

Stochastic Process

Stochastic Realization

Jump Time

Kolmogorov Extension Theorem

Clarity

Introducing Clarity and Perceivability