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Probability Terms and Definitions


Conditional Probability & Independence(3)
Continuous Distributions(2)
Discrete Distributions(7)
Distribution Functions(3)
Foundations(8)
Measures(8)
Multivariate Probability(11)
Random Variables(5)
Set Operations(6)
Transformations(3)
Visual Tools(1)
BBernoulli DistributionDiscrete DistributionsBBernoulli ExperimentRandom VariablesBBinomial DistributionDiscrete DistributionsBBivariate Random VariableMultivariate ProbabilityCComplement of a SetSet OperationsCConditional ExpectationMeasuresCConditional ProbabilityConditional Probability & IndependenceCConditional Probability Density FunctionMultivariate ProbabilityCConditional Probability Mass FunctionMultivariate ProbabilityCConditional VarianceMeasuresCContinuous Random VariableRandom VariablesCCorrelation CoefficientMeasuresCCovarianceMeasuresCCumulative Distribution FunctionDistribution FunctionsDDiscrete Random VariableRandom VariablesDDiscrete Uniform DistributionDiscrete DistributionsDDisjoint SetsSet OperationsEElementary EventFoundationsEEqually Likely EventsFoundationsEEventFoundationsEExpected ValueMeasuresEExponential DistributionContinuous DistributionsFFunction of a Random VariableTransformationsGGeometric DistributionDiscrete DistributionsHHypergeometric DistributionDiscrete DistributionsIIndependent EventsConditional Probability & IndependenceIIndependent Random VariablesMultivariate ProbabilityIIntersection of SetsSet OperationsJJoint Cumulative Distribution FunctionMultivariate ProbabilityJJoint Probability Density FunctionMultivariate ProbabilityJJoint Probability Mass FunctionMultivariate ProbabilityMMarginal DistributionMultivariate ProbabilityMMoment Generating FunctionTransformationsMMoment of a Random VariableMeasuresMMutual ExclusivenessConditional Probability & IndependenceNN-Variate Random VariablesMultivariate ProbabilityNNegative Binomial DistributionDiscrete DistributionsNNormal DistributionContinuous DistributionsNNull SetSet OperationsOOrthogonal Random VariablesMultivariate ProbabilityPPDF of a Transformed VariableTransformationsPPoisson DistributionDiscrete DistributionsPProbabilityFoundationsPProbability Density FunctionDistribution FunctionsPProbability Mass FunctionDistribution FunctionsPProbability MeasureFoundationsPProbability TreeVisual ToolsRRandom ExperimentFoundationsRRandom VariableRandom VariablesRRelative FrequencyFoundationsSSample SpaceFoundationsSSequence of Bernoulli TrialsRandom VariablesSStandard DeviationMeasuresUUncorrelated Random VariablesMultivariate ProbabilityUUnion of SetsSet OperationsVVarianceMeasuresVVenn DiagramSet Operations
57 of 57 terms
Conditional Probability & Independence
Conditional ProbabilityIndependent EventsMutual Exclusiveness
Continuous Distributions
Exponential DistributionNormal Distribution
Discrete Distributions
Negative Binomial DistributionBernoulli DistributionBinomial DistributionPoisson DistributionDiscrete Uniform DistributionGeometric DistributionHypergeometric Distribution
Distribution Functions
Cumulative Distribution FunctionProbability Mass FunctionProbability Density Function
Foundations
ProbabilityRandom ExperimentSample SpaceEventElementary EventRelative FrequencyProbability MeasureEqually Likely Events
Measures
Expected ValueVarianceStandard DeviationCovarianceCorrelation CoefficientConditional ExpectationConditional VarianceMoment of a Random Variable
Multivariate Probability
Bivariate Random VariableN-Variate Random VariablesIndependent Random VariablesOrthogonal Random VariablesUncorrelated Random VariablesMarginal DistributionJoint Cumulative Distribution FunctionJoint Probability Mass FunctionJoint Probability Density FunctionConditional Probability Mass FunctionConditional Probability Density Function
Random Variables
Bernoulli ExperimentSequence of Bernoulli TrialsRandom VariableDiscrete Random VariableContinuous Random Variable
Set Operations
Venn DiagramNull SetUnion of SetsIntersection of SetsDisjoint SetsComplement of a Set
Transformations
Function of a Random VariablePDF of a Transformed VariableMoment Generating Function
Visual Tools
Probability Tree

57 terms

Discrete Distributions

(7 items)

Negative Binomial Distribution

A discrete distribution counting the number of trials needed to achieve a fixed number rr of successes in a sequence of independent Bernoulli trials with success probability pp.
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intuitionnotationrelated concepts
The negative binomial distribution answers "how many trials until rr successes?" It generalizes the geometric distribution, which is the special case r=1r = 1.
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Bernoulli Distribution

A discrete distribution for a single trial with two outcomes: P(X=1)=pP(X = 1) = p and P(X=0)=1−pP(X = 0) = 1 - p.
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intuitionpropertiesrelated concepts
The Bernoulli distribution models a single yes/no trial. It is the simplest discrete distribution and the building block for the binomial, geometric, and negative binomial distributions.
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Binomial Distribution

A discrete distribution counting the number of successes in nn independent Bernoulli trials, each with success probability pp.
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intuitionnotationrelated concepts
The binomial distribution answers "how many successes?" in a fixed number of independent trials. It requires a fixed number of trials, constant probability, and independence.
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Poisson Distribution

A discrete distribution modelling the number of events occurring in a fixed interval, where events happen independently at a constant average rate λ\lambda.
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intuitionnotationrelated concepts
The Poisson distribution counts rare events in a fixed window of time or space. It applies when events occur independently and the average rate is known.
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Discrete Uniform Distribution

A discrete distribution where each of nn possible values has equal probability 1/n1/n.
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intuitionnotationrelated concepts
The discrete uniform distribution applies when every outcome is equally likely — the probability counterpart of "fair." A fair die is the classic example.
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Geometric Distribution

A discrete distribution counting the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials with success probability pp.
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intuitionnotationrelated concepts
The geometric distribution answers "how long until the first success?" It is the discrete analogue of the exponential distribution and shares its memoryless property.
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Hypergeometric Distribution

A discrete distribution describing the number of successes in nn draws without replacement from a finite population containing KK successes and N−KN - K failures.
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intuitionnotationrelated concepts
The hypergeometric distribution is used when sampling without replacement. Unlike the binomial, the probability of success changes from draw to draw because the population is finite.
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Foundations

(8 items)

Probability

A function PP that assigns to each event AA in a sample space a real number P(A)∈[0,1]P(A) \in [0, 1] satisfying the probability axioms.
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intuitionnotationrelated concepts
Probability quantifies how likely an event is to occur. A value of 0 means impossible, 1 means certain, and values in between reflect degrees of likelihood.
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Random Experiment

A process or action whose outcome cannot be predicted with certainty before it is performed.
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intuitionexamplesrelated concepts
A random experiment is any procedure that produces an unpredictable result. Rolling a die, drawing a card, or measuring a physical quantity are all random experiments.
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Sample Space

Ω={ω1,ω2,…}\Omega = \{\omega_1, \omega_2, \ldots\} — the set of all possible outcomes of a random experiment.
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intuitionnotationrelated concepts
The sample space is the complete list of everything that can happen. Every probability question begins by identifying this set, because events and probabilities are defined relative to it.
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Event

A⊆ΩA \subseteq \Omega — a subset of the sample space.
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intuitionexamplesrelated concepts
An event is a collection of outcomes we are interested in. It can contain one outcome, several outcomes, or even all outcomes. Probabilities are assigned to events, not to individual outcomes directly.
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Elementary Event

An event consisting of exactly one outcome: {ω}\{\omega\} where ω∈Ω\omega \in \Omega.
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intuitionrelated concepts
An elementary event is the simplest possible event — a single outcome from the sample space that cannot be broken down further.
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Relative Frequency

fn(A)=number of times A occursnf_n(A) = \frac{\text{number of times } A \text{ occurs}}{n} where nn is the total number of trials.
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intuitionrelated concepts
Relative frequency is the proportion of times an event occurs in repeated experiments. As the number of trials grows, relative frequency tends to stabilize near the true probability of the event.
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Probability Measure

A function P:F→[0,1]P: \mathcal{F} \to [0,1] defined on a collection of events, satisfying non-negativity, normalization (P(Ω)=1P(\Omega) = 1), and countable additivity for disjoint events.
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intuitionpropertiesrelated concepts
A probability measure is the formal rule that assigns a number between 0 and 1 to every event in a way that is internally consistent. It is the mathematical object that makes probability rigorous.
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Equally Likely Events

Events A1,A2,…,AnA_1, A_2, \ldots, A_n are equally likely when P(A1)=P(A2)=⋯=P(An)P(A_1) = P(A_2) = \cdots = P(A_n).
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intuitionexamplesrelated concepts
When all outcomes in a finite sample space have the same probability, they are equally likely. In this case probability reduces to counting: P(A)=∣A∣/∣Ω∣P(A) = |A| / |\Omega|.
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Conditional Probability & Independence

(3 items)

Conditional Probability

P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}, defined when P(B)>0P(B) > 0.
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intuitionnotationrelated concepts
Conditional probability measures the likelihood of an event given that another event has already occurred. It restricts attention to a smaller part of the sample space.
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Independent Events

Events AA and BB are independent if and only if P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).
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intuitioncommon errorsrelated concepts
Two events are independent when the occurrence of one provides no information about the other. Knowing that BB happened does not change the probability of AA.
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Mutual Exclusiveness

Events AA and BB are mutually exclusive if A∩B=∅A \cap B = \emptyset.
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intuitionpropertiesrelated concepts
Mutually exclusive events cannot happen at the same time. If one occurs, the other is automatically ruled out.
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Random Variables

(5 items)

Bernoulli Experiment

A random experiment with exactly two possible outcomes, conventionally called success (SS) and failure (FF).
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intuitionexamplesrelated concepts
A Bernoulli experiment is the simplest random experiment: something either happens or it does not. It is the building block for more complex models like the binomial distribution.
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Sequence of Bernoulli Trials

A sequence of independent Bernoulli experiments, each with the same success probability pp.
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intuitionrelated concepts
Repeating the same yes/no experiment independently under identical conditions. The number of successes, the trial of first success, and similar quantities each give rise to a named distribution.
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Random Variable

X:Ω→RX: \Omega \to \mathbb{R} — a function that assigns a real number to each outcome in the sample space.
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intuitionnotationrelated concepts
A random variable translates outcomes of a random experiment into numbers. This numerical representation makes it possible to compute averages, measure spread, and define distributions.
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Discrete Random Variable

A random variable whose set of possible values is finite or countably infinite.
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intuitionexamplesrelated concepts
A discrete random variable takes on isolated, separated values that can be listed — even if the list is infinite. Its probability distribution is described by a probability mass function.
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Continuous Random Variable

A random variable whose set of possible values forms an interval or union of intervals on the real line.
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intuitionexamplesrelated concepts
A continuous random variable can take any value within a range. Probability is spread smoothly rather than concentrated at individual points, and is described by a probability density function.
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Distribution Functions

(3 items)

Cumulative Distribution Function

FX(x)=P(X≤x)F_X(x) = P(X \le x) for all x∈Rx \in \mathbb{R}.
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intuitionpropertiesrelated concepts
The CDF tracks how much probability has accumulated up to each value. It answers "how likely is the random variable to be at most xx?" and works for any type of distribution.
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Probability Mass Function

pX(x)=P(X=x)p_X(x) = P(X = x) — the probability that a discrete random variable XX takes the value xx.
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intuitionpropertiesrelated concepts
The PMF assigns a probability to each individual value a discrete random variable can take. All values are non-negative and their sum equals 1.
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Probability Density Function

A function fX(x)≥0f_X(x) \ge 0 such that P(a≤X≤b)=∫abfX(x) dxP(a \le X \le b) = \int_a^b f_X(x)\,dx and ∫−∞∞fX(x) dx=1\int_{-\infty}^{\infty} f_X(x)\,dx = 1.
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intuitioncommon errorsrelated concepts
The PDF describes how densely probability is spread across values of a continuous random variable. Its value at a point is not a probability — probabilities come from integrating the PDF over intervals.
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Measures

(8 items)

Expected Value

E[X]=∑xx⋅pX(x)E[X] = \sum_x x \cdot p_X(x) (discrete) or E[X]=∫−∞∞x⋅fX(x) dxE[X] = \int_{-\infty}^{\infty} x \cdot f_X(x)\,dx (continuous).
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intuitionpropertiesrelated concepts
The expected value is the long-run average of a random variable over many repetitions. It represents the center of mass of the distribution.
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Variance

Var⁡(X)=E[(X−μ)2]\operatorname{Var}(X) = E[(X - \mu)^2] where μ=E[X]\mu = E[X].
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intuitionpropertiesrelated concepts
Variance measures how spread out a random variable's values are around its expected value. A larger variance means outcomes are more dispersed.
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Standard Deviation

σX=Var⁡(X)\sigma_X = \sqrt{\operatorname{Var}(X)}
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intuitionrelated concepts
Standard deviation is the square root of variance. It measures spread in the same units as the random variable, making it easier to interpret than variance.
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Covariance

Cov⁡(X,Y)=E[(X−E[X])(Y−E[Y])]\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]
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intuitionpropertiesrelated concepts
Covariance measures how two random variables move together. Positive covariance means they tend to increase together; negative means one tends to increase when the other decreases.
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Correlation Coefficient

ρXY=Cov⁡(X,Y)σX⋅σY\rho_{XY} = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}, where σX,σY>0\sigma_X, \sigma_Y > 0.
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intuitionpropertiesrelated concepts
The correlation coefficient normalizes covariance to a scale between −1-1 and 11. It measures the strength and direction of the linear relationship between two random variables.
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Conditional Expectation

E[X∣Y=y]=∑xx⋅P(X=x∣Y=y)E[X \mid Y = y] = \sum_x x \cdot P(X = x \mid Y = y) (discrete) or E[X∣Y=y]=∫x⋅fX∣Y(x∣y) dxE[X \mid Y = y] = \int x \cdot f_{X|Y}(x \mid y)\,dx (continuous).
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intuitionrelated concepts
Conditional expectation is the expected value of one random variable given that another takes a specific value. It adjusts the average to account for known information.
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Conditional Variance

Var⁡(X∣Y=y)=E[(X−E[X∣Y=y])2∣Y=y]\operatorname{Var}(X \mid Y = y) = E[(X - E[X \mid Y = y])^2 \mid Y = y]
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intuitionrelated concepts
Conditional variance measures the spread of one random variable around its conditional mean, given a specific value of another variable.
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Moment of a Random Variable

The kk-th moment of XX about the origin is E[Xk]E[X^k]. The kk-th central moment is E[(X−μ)k]E[(X - \mu)^k].
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intuitionrelated concepts
Moments are numerical summaries of a distribution's shape. The first moment is the mean, the second central moment is the variance, and higher moments capture skewness and tail behavior.
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Continuous Distributions

(2 items)

Exponential Distribution

A continuous distribution describing the time between events in a process where events occur independently at a constant rate λ\lambda.
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intuitionnotationrelated concepts
The exponential distribution models waiting time — how long until the next event. Its defining feature is the memoryless property: the probability of waiting longer does not depend on how long you have already waited.
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Normal Distribution

A continuous distribution characterized by a symmetric, bell-shaped curve, fully determined by its mean μ\mu and variance σ2\sigma^2.
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intuitionnotationrelated concepts
The normal distribution appears when many small independent effects combine. Its bell curve is symmetric about the mean, with probability decreasing smoothly in both directions.
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Multivariate Probability

(11 items)

Bivariate Random Variable

A pair of random variables (X,Y)(X, Y) defined on the same sample space, considered jointly.
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intuitionrelated concepts
A bivariate random variable treats two measurements taken from the same experiment as a single object. Their joint behavior — how they relate, co-occur, or depend on each other — is captured by a joint distribution.
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N-Variate Random Variables

A vector (X1,X2,…,Xn)(X_1, X_2, \ldots, X_n) of nn random variables defined on the same sample space.
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intuitionrelated concepts
An extension of bivariate random variables to any number of components. The joint distribution describes how all nn variables behave together.
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Independent Random Variables

Random variables XX and YY are independent if P(X≤x,Y≤y)=P(X≤x)⋅P(Y≤y)P(X \le x, Y \le y) = P(X \le x) \cdot P(Y \le y) for all x,yx, y.
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intuitionpropertiesrelated concepts
Independent random variables carry no information about each other. Knowing the value of one does not change the distribution of the other.
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Orthogonal Random Variables

Random variables XX and YY are orthogonal if E[XY]=0E[XY] = 0.
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intuitionrelated concepts
Orthogonality is an algebraic condition on the product of two random variables. It is weaker than independence and does not imply zero covariance unless the means are zero.
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Uncorrelated Random Variables

Random variables XX and YY are uncorrelated if Cov⁡(X,Y)=0\operatorname{Cov}(X, Y) = 0, equivalently E[XY]=E[X]E[Y]E[XY] = E[X]E[Y].
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intuitioncommon errorsrelated concepts
Uncorrelated means no linear relationship between two variables. Independent random variables are always uncorrelated, but uncorrelated variables can still be dependent through nonlinear relationships.
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Marginal Distribution

The distribution of one random variable obtained from a joint distribution by summing (discrete) or integrating (continuous) over all values of the other variable(s).
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intuitionrelated concepts
A marginal distribution extracts the standalone behavior of one variable from a joint distribution, ignoring the other variables. In a contingency table, marginals appear as row or column totals.
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Joint Cumulative Distribution Function

FX,Y(x,y)=P(X≤x,Y≤y)F_{X,Y}(x, y) = P(X \le x, Y \le y) for all x,y∈Rx, y \in \mathbb{R}.
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intuitionrelated concepts
The joint CDF extends the cumulative distribution function to multiple random variables. It gives the probability that all variables simultaneously fall below their respective thresholds.
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Joint Probability Mass Function

pX,Y(x,y)=P(X=x,Y=y)p_{X,Y}(x, y) = P(X = x, Y = y) for discrete random variables XX and YY.
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intuitionpropertiesrelated concepts
The joint PMF assigns a probability to each specific combination of values two discrete random variables can take simultaneously.
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Joint Probability Density Function

A function fX,Y(x,y)≥0f_{X,Y}(x,y) \ge 0 such that P((X,Y)∈A)=∬AfX,Y(x,y) dx dyP((X,Y) \in A) = \iint_A f_{X,Y}(x,y)\,dx\,dy for any region AA.
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intuitionrelated concepts
The joint PDF describes how probability density is spread over the plane for two continuous random variables. Probabilities are obtained by integrating over regions, not by evaluating at points.
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Conditional Probability Mass Function

pX∣Y(x∣y)=pX,Y(x,y)pY(y)p_{X|Y}(x \mid y) = \frac{p_{X,Y}(x, y)}{p_Y(y)}, defined when pY(y)>0p_Y(y) > 0.
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intuitionrelated concepts
The conditional PMF gives the probability distribution of one discrete random variable when another discrete random variable is known to take a specific value.
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Conditional Probability Density Function

fX∣Y(x∣y)=fX,Y(x,y)fY(y)f_{X|Y}(x \mid y) = \frac{f_{X,Y}(x, y)}{f_Y(y)}, defined when fY(y)>0f_Y(y) > 0.
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intuitionrelated concepts
The conditional PDF describes the density of one continuous random variable when another continuous random variable is known to take a specific value.
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Transformations

(3 items)

Function of a Random Variable

Given a random variable XX and a function gg, Y=g(X)Y = g(X) defines a new random variable whose distribution is determined by the distribution of XX and the function gg.
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intuitionrelated concepts
Applying a function to a random variable produces a new random variable. The challenge is determining the distribution of the result from the distribution of the original.
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PDF of a Transformed Variable

If Y=g(X)Y = g(X) where gg is monotone and differentiable, then fY(y)=fX(g−1(y))⋅∣ddyg−1(y)∣f_Y(y) = f_X(g^{-1}(y)) \cdot |\frac{d}{dy} g^{-1}(y)|.
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intuitionrelated concepts
When a continuous random variable is transformed, its density changes. The change-of-variables formula accounts for both the mapping of values and the stretching or compression of the density.
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Moment Generating Function

MX(t)=E[etX]M_X(t) = E[e^{tX}], defined for real values of tt where the expectation exists.
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intuitionpropertiesrelated concepts
The moment generating function encodes all moments of a random variable in a single function. The kk-th moment is obtained by differentiating kk times and evaluating at t=0t = 0.
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Set Operations

(6 items)

Venn Diagram

A graphical representation using overlapping circles to depict sets (events) and their relationships within a sample space.
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intuitionrelated concepts
Venn diagrams visualize how events overlap, combine, or exclude each other. They make set operations like union, intersection, and complement immediately visible.
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Null Set

∅\emptyset — the set containing no elements, representing an impossible event in probability.
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intuitionrelated concepts
The null set is the event that can never occur. Its probability is always zero: P(∅)=0P(\emptyset) = 0.
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Union of Sets

A∪B={ω:ω∈A or ω∈B}A \cup B = \{\omega : \omega \in A \text{ or } \omega \in B\} — the event that at least one of AA or BB occurs.
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intuitionrelated concepts
The union combines two events into one that occurs whenever either event (or both) occurs.
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Intersection of Sets

A∩B={ω:ω∈A and ω∈B}A \cap B = \{\omega : \omega \in A \text{ and } \omega \in B\} — the event that both AA and BB occur simultaneously.
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intuitionrelated concepts
The intersection is the event where both conditions are met at the same time.
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Disjoint Sets

Sets AA and BB are disjoint if A∩B=∅A \cap B = \emptyset — they share no common elements.
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intuitionrelated concepts
Disjoint sets have no overlap. In probability, disjoint events are mutually exclusive — they cannot both occur in the same trial.
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Complement of a Set

Ac={ω∈Ω:ω∉A}A^c = \{\omega \in \Omega : \omega \notin A\} — all outcomes in the sample space that are not in AA.
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intuitionrelated concepts
The complement of an event is everything that can happen except that event. P(Ac)=1−P(A)P(A^c) = 1 - P(A).
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Visual Tools

(1 items)

Probability Tree

A branching diagram where each node represents a stage of a sequential random process, branches represent possible outcomes, and branch labels show conditional probabilities.
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intuitionrelated concepts
A probability tree lays out a multi-stage experiment visually. Multiplying along a path gives joint probabilities; summing across paths reaching the same outcome gives marginal probabilities.
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