Random vector

A random vector is a measurable function from a probability space into a multidimensional value space, assigning a vector to each outcome.

Probability space: A measure space whose total measure is one, used to model random outcomes and their probabilities. It supplies the outcomes and probability measure on which a random vector is defined.
Joint distribution: The probability distribution of two or more random variables considered together. It records the probabilities of regions in the random vector's value space.
Multivariate normal distribution: A probability distribution on real vectors whose every linear combination is normally distributed. It is a central model for random vectors described by a mean and covariance matrix.
Scalar random variable: A measurable function from a probability space into a one-dimensional value space. It carries one coordinate of randomness rather than a multidimensional outcome.
Measurable function: A function whose inverse images of measurable sets are measurable. Measurability makes probabilities of vector-valued events well-defined.
Marginal distribution: The distribution of a subset of variables obtained from a joint distribution. Each coordinate or coordinate subset has a marginal derived from the vector's law.
Multivariate statistics: Statistical methods for analyzing multiple variables jointly. Random vectors provide the mathematical observations underlying this field.
Random element: A measurable function from a probability space into a measurable space. This broader formulation allows value spaces beyond finite-dimensional vector spaces.
Random variable: A measurable function from a probability space into a measurable value space, often the real numbers. A random vector generalizes a scalar random variable to multiple coordinates.
Cumulative distribution function: A function giving the probability that a random variable is at most a specified value. Its multivariate form describes probabilities across coordinate-wise lower regions.

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