A measure of the likelihood that a specific event will occur, expressed as a value between 0 and 1.
Random Experiment
Definition:
A process or action that produces uncertain outcomes, such as rolling a die or tossing a coin.
Sample Space
Definition:
The set of all possible outcomes of a random experiment.
Event
Definition:
A subset of the sample space, representing one or more outcomes of interest in a random experiment.
Elementary Event
Definition:
A single outcome from the sample space that cannot be decomposed further.
Set Operations
Set Operations
Definition:
Mathematical operations (like union, intersection, and complement) used to combine or relate sets.
Null Set
Definition:
A set with no elements, representing an impossible event in probability.
Union Of Sets
Definition:
A set that contains all elements from either or both of the sets being combined.
Intersection Of Sets
Definition:
A set containing only the elements that are common to all sets being compared.
Disjoint Sets
Definition:
Sets that have no elements in common.
Venn Diagram
Definition:
A graphical representation of sets and their relationships using overlapping circles.
Complement Of A Set
Definition:
The set of elements in the sample space that are not in the given set.
De Morgan's Laws
Definition:
Mathematical rules relating the complement of a union or intersection of sets to the intersection or union of their complements.
Basic Probability
Relative Frequency
Definition:
The ratio of the number of times an event occurs to the total number of trials or observations.
Probability Measure
Definition:
A function assigning a probability value to events within a sample space while satisfying certain axioms.
Axiomatic Probability
Definition:
A formal framework defining probability using a set of axioms ensuring logical consistency.
Elementary Properties Of Probability
Definition:
Basic rules of probability, including values between 0 and 1, and relationships between events like union and intersection.
Equally Likely Events
Definition:
Events with the same probability of occurrence in an experiment.
Independent Events
Definition:
Events whose occurrences are not influenced by each other.
Mutual Exclusiveness
Definition:
A condition where two or more events cannot occur simultaneously.
Conditional Probability
Conditional Probability
Definition:
The probability of one event occurring given that another event has occurred.
Bayes' Rule
Definition:
A formula to update the probability of an event based on new information about related events.
Total Probability
Definition:
A theorem that expresses the probability of an event as the sum of probabilities of it occurring under different conditions.
Conditional Probability Mass Function
Definition:
The probability distribution of a discrete random variable given that another discrete random variable takes a specific value.
Conditional Probability Density Function
Definition:
The probability density of a continuous random variable given that another continuous random variable takes a specific value.
Inequalities
Bonferroni's Inequality
Definition:
A relationship providing bounds for the probability of the union of events.
Boole's Inequality
Definition:
An upper bound on the probability of the union of several events.
Chebyshev's Inequality
Definition:
A statistical inequality providing a bound on the probability that a random variable deviates from its mean.
Markov Inequality
Definition:
An inequality bounding the probability of a non-negative random variable exceeding a given value.
Random Variables
Bernoulli Experiment
Definition:
A random experiment with exactly two possible outcomes, typically labeled as success and failure.
Sequence Of Bernoulli Trials
Definition:
Repeated independent Bernoulli experiments where the probability of success remains constant across trials.
Random Variable
Definition:
A function that assigns numerical values to outcomes in a sample space, enabling the study of probabilities of events.
Discrete Random Variable
Definition:
A random variable with a countable set of possible values.
Continuous Random Variable
Definition:
A random variable with an uncountable set of values, typically forming an interval on the real number line.
Independent Random Variables
Definition:
Random variables whose outcomes do not influence each other's probabilities.
Orthogonal Random Variables
Definition:
Random variables with zero covariance, indicating no linear relationship.
Uncorrelated Random Variables
Definition:
Random variables with zero correlation coefficient, implying no linear relationship.
Distribution Functions
Cumulative Distribution Function
Definition:
A function that gives the probability that a random variable is less than or equal to a given value.
Probability Mass Function
Definition:
A function that specifies the probability of each possible value for a discrete random variable.
Probability Density Function
Definition:
A function describing the likelihood of a continuous random variable taking on a specific value within an interval.
Probability Density Function Of A Transformed Variable
Definition:
The function that describes the distribution of probabilities for a random variable obtained through transformation.
Statistical Measures
Expected Value
Definition:
The weighted average of all possible values of a random variable, reflecting its long-term average.
Variance
Definition:
A measure of the spread or dispersion of a random variable, calculated as the average squared deviation from the mean.
Standard Deviation
Definition:
The square root of the variance, providing a measure of spread in the same units as the random variable.
Covariance
Definition:
A measure of how two random variables change together, indicating the direction of their relationship.
Correlation Coefficient
Definition:
A normalized measure of the linear relationship between two variables, ranging from -1 to 1.
Conditional Expectation
Definition:
The expected value of one random variable given the value of another random variable.
Conditional Variance
Definition:
The variance of a random variable given that another random variable takes on a specific value.
Moment Of A Random Variable
Definition:
A quantitative measure of the shape of the variable's probability distribution, derived as the expected value of its powers.
Probability Distributions
Bernoulli Distribution
Definition:
A discrete distribution describing the outcome of a single trial with two possible outcomes, success and failure.
Binomial Distribution
Definition:
A discrete distribution of the number of successes in a fixed number of independent Bernoulli trials.
Poisson Distribution
Definition:
A discrete distribution modeling the number of events occurring in a fixed interval, assuming events occur independently.
Uniform Distribution
Definition:
A distribution where all outcomes in a specified range are equally likely.
Exponential Distribution
Definition:
A continuous distribution describing the time between events in a Poisson process, with the memoryless property.
Normal Distribution
Definition:
A continuous distribution characterized by its bell-shaped curve, symmetric about the mean.
Rayleigh Distribution
Definition:
A continuous distribution often used in signal processing, describing the magnitude of a vector in two dimensions.
Gamma Distribution
Definition:
A continuous distribution that generalizes the exponential distribution, used in reliability and queuing models.
Hypergeometric Distribution
Definition:
A discrete distribution describing probabilities in draws without replacement from a finite population.
Geometric Distribution
Definition:
A discrete distribution representing the number of trials needed to get the first success in repeated Bernoulli trials.
Multinomial Distribution
Definition:
A generalization of the binomial distribution for more than two possible outcomes in each trial.
Bivariate Normal Distribution
Definition:
A distribution where two continuous random variables are jointly normally distributed.
N-Variate Normal Distribution
Definition:
A generalization of the bivariate normal distribution to more than two dimensions.
Advanced Concepts
Markov Property
Definition:
The memoryless property where the future state depends only on the current state and not on past states.
Central Limit Theorem
Definition:
A theorem stating that the sum of many independent random variables tends toward a normal distribution, regardless of the original distributions.
Central Limit Theorem For N-Variate
Definition:
A theorem stating that the sum of multiple independent random variables approximates a multivariate normal distribution under certain conditions.
Function Of A Random Variable
Definition:
A rule that assigns a new random variable based on a transformation of an existing one, typically denoted as Y=g(X).
Moment Generating Function
Definition:
A function used to describe all moments of a random variable, defined as the expected value of e^(tX) for a real parameter t.
Characteristic Function
Definition:
The Fourier transform of a probability distribution, useful for studying the properties and behaviors of random variables.
Weak Law Of Large Numbers
Definition:
A theorem stating that the sample mean of independent, identically distributed random variables converges in probability to their true mean as the sample size increases.
Strong Law Of Large Numbers
Definition:
A theorem that states the sample mean almost surely converges to the true mean as the sample size grows infinitely large.
Central Limit Theorem
Definition:
A fundamental result in probability theory stating that the sum of a large number of independent, identically distributed random variables will be approximately normally distributed.
Multivariate Probability
Bivariate Random Variable
Definition:
A pair of random variables considered together, forming a two-dimensional vector defined on the same sample space.
Joint Cumulative Distribution Function
Definition:
A function that gives the probability that two random variables simultaneously take on values less than or equal to specific values.
Marginal Distribution
Definition:
The probability distribution of one random variable obtained by summing or integrating out the other variable in a joint distribution.
Joint Probability Mass Function
Definition:
A function giving the probability that two discrete random variables simultaneously take on specific values.
Joint Probability Density Function
Definition:
A function representing the probability density of two continuous random variables taking on specific values.
N-Variate Random Variables
Definition:
A set of multiple random variables considered as a vector, defining a multi-dimensional space.