Random Variables and Probability Measures

A set of vocabulary flashcards focused on key concepts related to random variables and probability measures.

Discrete Random Variable

A variable that can take on a finite number of distinct values, such as heads or tails from a coin toss.

Continuous Random Variable

A variable that can take any value within a given range, often represented by a probability density function.

Probability Density Function (PDF)

A function that describes the likelihood of a continuous random variable to take on a particular value.

Cumulative Distribution Function (CDF)

A function used to describe the probability that a random variable takes on a value less than or equal to a specific value.

Random Vector

A vector whose components are random variables, which can be discrete or continuous.

Sigma Field

A collection of sets that is closed under complementation and countable unions, used in probability theory.

Probability Measure

A function that assigns a probability to subsets of a given sample space, ensuring that probabilities are between 0 and 1.

Measurable Function

A function for which the pre-image of any Borel set is a measurable set, leading to a well-defined integral.

Radon-Nikodym Density Theorem

A theorem that states the existence of a density function for probability measures on mixed random vectors.

Support of a Random Variable

The set of values that a random variable can take with non-zero probability.

Expectation

The average value of a random variable, calculated as the integral of the variable weighted by its probability measure.

Concentration Inequality

An inequality that provides bounds on the probability that a random variable will deviate from its expected value.

Chebyshev's Inequality

An inequality that states that the probability of a random variable deviating from its mean by more than k standard deviations is less than or equal to 1/k^2.

Continuous Random Function

A function that maps inputs from a random vector space to a random output, preserving measurability in its first argument.

Probability Mass Function (PMF)

A function that gives the probability that a discrete random variable is exactly equal to some value.

Variance

A measure of the spread of a random variable's distribution around its mean, calculated as the expected value of the squared deviation from the mean, often denoted as Var(X) = E[(X - E[X])^2].

Standard Deviation

The square root of the variance, providing a measure of the typical deviation of values from the mean in the same units as the random variable, denoted as \sigma = \sqrt{Var(X)}.