Random Variable

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Random Variable

Definition (*)

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).

Proposition (3.1.5)

If $A\in\mathcal{F}$ , then for $X(\omega)=\mathbb{1}_{A}(\omega)$ , we have that $X$ is an rv
If $X,Y$ are rvs and $c$ is some constant, then $X+c,cX,XY,X+Y$ are rvs.
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 $rvs$ .

Proposition (3.1.8)

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

Beppo Levi Theorem

Change of Variable Formula

Fubini-Tonelli

Blackwell's Irrelevant Information Theorem

Controlled Markov Chain

Policy

Kalman Filter

Information Signal

LQG Teams

Radner Krainak Theorem

Bayesian Statistics

Likelihood

Maximal Likelihood Estimator

Monte Carlo Method

Redundancy

Optimal Bit Allocation

Continuous Memoryless Source

Differential Divergence

Differential Entropy

Estimation Error and differential entropy

Data Processing Inequality

Discrete Memoryless Source

Entropy

Joint Entropy

Mutual Information

Renyi Entropy

Source Entropy

Source

Closed-loop Predictor Coefficients

Difference Quantization

Linear Prediction

Wide Sense Stationary Process

Almost Sure Convergence

Central Limit Theorem

Convergence in Distribution

Convergence in Expectation

Existence of Sequences of Independent rvs

In Probability Convergence

Law of Large Numbers

Pointwise Convergence

Portmanteau's Theorem

Scheffé's Theorem

Skohorod's Theorem

Weak convergence

Cauchy-Schwartz Inequality

Conditional Expectation

Conditional Variance

Covariance

Expectation of a Function of a Random Variable

Expectation

Markov's Inequality

Moment

Orthogonal random variables

Variance

Conditional Independence

De Finetti's Theorem

Distribution

Independent

Random Variable

Random Vector

iid

σ(X)

(Cumulative) Distribution Function

Conditional probability function

Joint probability function

Marginal probability 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

Uniform Random Variable

Predicable Processes

Portfolio

Kolmogorov Extension Theorem

Doob's Upcrossing Inequality

Martingale Convergence Theorem

Dirac Distribution

Stochastic Process

Stochastic Realization

Jump Time

Continuous-time Gaussian process motion planning via probabilistic inference