Probability distribution

A mathematical description assigning probabilities to the possible values or outcomes of a random variable or random process. It may be discrete, continuous, or a mixture of both.

Random variable: A function that assigns a numerical value to each outcome in a random experiment. A distribution describes the probabilities associated with a random variable’s possible values.
Expected value: The probability-weighted average value of a random variable, when that average exists. It extracts a central summary directly from the distribution’s probabilities.
Binomial distribution: The distribution of the number of successes in a fixed number of independent trials with equal success probability. It models counts when trials share two outcomes and a constant success chance.
Joint probability distribution: A distribution describing the probabilities of combinations of values for multiple random variables. It extends a single-variable distribution to capture dependence among variables.
Probability measure: A function assigning probabilities to events in a sample space, satisfying the probability axioms. Distributions inherit their probabilities from a measure on the variable’s outcomes.
Variance: The expected squared deviation of a random variable from its mean. It measures how widely the distribution spreads around its expected value.
Poisson distribution: A discrete distribution for counts of events occurring at a constant average rate in a fixed interval. It models event counts and approximates binomial probabilities under suitable rare-event conditions.
Conditional probability distribution: A distribution of one or more random variables given specified information about other variables or events. It changes the probabilities being described by conditioning on known information.
Cumulative distribution function: A function giving the probability that a random variable is less than or equal to each value. It fully specifies a distribution and works for discrete, continuous, and mixed cases.
Moment-generating function: A function whose derivatives at zero, when they exist, give a random variable’s moments. It can characterize a distribution and make sums of independent variables easier to analyze.

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