Stat 5101 Lecture Slides: Deck 1 Probability and Expectation on Finite Sample Spaces

Vocabulary flashcards covering key terms related to probability and expectation on finite sample spaces, based on lecture slides.

Set (mathematical definition)

A collection of objects considered as one thing; objects are called elements.

Element (of a set)

An object within a set.

Subset

A set A is a subset of set S if every element of A is an element of S (A ⊂ S).

Empty Set

The only set that contains no elements, denoted by ∅.

Natural Numbers (N)

The set of non-negative integers: {0, 1, 2, …}.

Integers (Z)

The set of all whole numbers (including negative numbers and zero): {…, -2, -1, 0, 1, 2, …}.

Real Numbers (R)

The set of all rational and irrational numbers.

Set Builder Notation

A notation for defining a set by specifying a condition that its elements must satisfy, e.g., { x ∈ S : some condition on x }.

Interval

A set of real numbers between two specified values (endpoints).

Open Interval

An interval that does not include its endpoints, denoted (a, b) = { x ∈ R : a < x < b }.

Closed Interval

An interval that includes its endpoints, denoted [a, b] = { x ∈ R : a ≤ x ≤ b }.

Half-Open Interval

An interval that includes one endpoint but not the other, denoted (a, b] or [a, b).

Function

A rule that assigns to each point in its domain, exactly one point in its codomain. Also called maps, mappings, or transformations.

Domain (of a Function)

The set of all possible inputs for a function.

Codomain (of a Function)

The set that contains all possible outputs of a function.

Probability Model

Also called probability distribution, a mathematical description of a random phenomenon.

Probability Mass Function (PMF)

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

Sample Space

The set of all possible outcomes of a random experiment.

Outcome

An element of the sample space.

Finite Probability Model

A probability model with a finite sample space.

Bernoulli Distribution

A probability distribution on the sample space {0, 1}, denoted Ber(p) where p is the probability of success (1).

Statistical Model

A family of probability models.

Parameter (of a Statistical Model)

A value that determines a specific probability model within a statistical model.

Parameter Space

The set of all possible values of the parameter(s) in a statistical model.

Discrete Uniform Distribution

A distribution where all outcomes in the sample space have equal probability.

Support (of a distribution)

The set of values for which the PMF is strictly positive.

Event

A subset of the sample space.

Probability Measure

A function that assigns a probability to each event.

Random Variable

A real-valued function on the sample space.

Expectation

The expected value of a random variable, calculated as the sum of each possible value times its probability.

Expectation Operator

A function that maps random variables to their expected values.

Cartesian Product

The set of all ordered pairs (or n-tuples) where the first element comes from the first set, the second from the second set, and so on.

Set Difference

The set of all elements of A that are not in B, denoted A \ B.

Indicator Function

A function that indicates whether an element is a member of a set (1) or not (0).

Frequentism

A theory that probabilities are objective facts measured in infinite sequences of repetitions.

Subjectivism

A theory that probabilities are personal reflections of uncertainty.

Formalism

The main stream theory that cares about the form of mathematics arguments.

Independence (stochastic)

Random variables X1, …, Xn are independent if the PMF of the random vector (X1, …, Xn) is the product of the PMFs of the component random variables.

n Factorial (n!)

The product of all positive integers less than or equal to n.

Combinations

The number of ways to choose k things from n distinct things where order doesn't matter.

Permutations

The number of ways to arrange k things chosen from n distinct things.

Binomial Theorem

A theorem that states the expansion of (a + b)^n in terms of binomial coefficients.

Binomial Distribution

The probability distribution of the number of successes in a fixed number of independent Bernoulli trials.

IID

independent and identically distributed