STATE 05 · REFERENCE

Glossary of terms

A compact restatement of the vocabulary used across the reference.

The definitions below are deliberately brief. Their purpose is to recall a term to a reader who has already met it, not to substitute for the fuller treatment on the pages where the term belongs. Each entry is followed by a pointer to the state in which the idea is developed at length.

Random variable: A measurable function that assigns a real number to each outcome of an experiment. Its law is the measure induced on the real line. The most general starting point of the subject.
Probability measure: A function that assigns to each measurable event a number between zero and one, obeying the axioms of countable additivity and unit total mass.
Density function: The local weight of an absolutely continuous law. Its integral over a region returns the probability of that region, and its integral over the whole line equals one.
Cumulative distribution function: The function that returns the probability of the values not exceeding a given point. It exists for every law, is non-decreasing, and rises from zero to one.
Expectation: The probability-weighted average of a variable, written as an integral against its law. The operation is linear, and it extends to functions of the variable by the same integral.
Variance: The expectation of the squared deviation of a variable from its mean. It measures spread and equals the second central moment.
Moment: The expectation of a power of a variable. The moments, taken together, summarise the shape of a law, including its spread, its asymmetry and the weight of its tails.
Covariance: The expectation of the product of the deviations of two variables from their means. It measures the tendency of the pair to move together, and it depends on the units of measurement.
Correlation: The covariance divided by the two standard deviations. It is a pure number between minus one and one, and it measures linear association without regard to units.
Stationarity: The property of a series whose probabilistic character does not shift as the index advances. Weak stationarity constrains the first two moments; strict stationarity constrains every finite-dimensional law.
Autocorrelation: The correlation between the values of a series separated by a stated lag. Its pattern as the lag grows is the signature of the series.
Autocovariance: The covariance between the values of a series at two indices. For a stationary series it depends only on the separation of the indices.
Spectral density: The Fourier transform of the autocovariance function. It reports how the variance of a series is distributed across frequencies, and it integrates to the total variance.
Transition matrix: A square table whose entry in row i and column j is the probability of moving from state i to state j in one step of a Markov chain. Every row sums to one.
Markov property: The statement that the future of a series, given its present value, does not depend on its earlier past.
Estimator: A function of the observations intended to approximate an unknown parameter. It is itself a random variable, and it is assessed through its bias, its variance and its consistency.
Likelihood: The joint density of the observed values, regarded as a function of the unknown parameter. Its maximum supplies the maximum likelihood estimate.
Confidence interval: A random interval, built from the data, that covers the unknown parameter with a stated long-run frequency over repeated constructions.
p-value: The probability, under a stated null hypothesis, of obtaining a result at least as extreme as the one observed. It is a statement about the data given the null.

The list is not exhaustive, and the pages of the reference supply the context that these short entries necessarily omit. A reader who wishes to begin again at the start may return to the home page, where the five states of the reference are set out in turn.