Do the moments of a distribution always determine it?

Not always; under growth conditions on the moments they do, but some distributions, such as the log-normal, share every moment with distinct distributions, so the moment sequence can fail to pin down the law.

Why introduce cumulants alongside moments?

Cumulants add over independent random variables, so they behave more simply for sums than moments do; the second cumulant is the variance and higher cumulants measure departures from normality, all of which vanish above order two for the normal distribution.

Transformations and Moments

Functions of random variables have distributions of their own, found by change-of-variables formulas, and moments and their generating functions summarize a distribution through its mean, variance, and higher-order shape.

Definition

A transformation of a random variable is a measurable function of it whose distribution is obtained by pushing forward the original law, and moments are the expectations of powers of a random variable that summarize the location, spread, and shape of its distribution.

Scope

The topic covers the distribution of functions of one or several random variables by the change-of-variables and Jacobian formulas, moments and central moments, variance and covariance, the moment and cumulant generating functions, the relations among moments, cumulants, skewness, and kurtosis, and the moment problem of when moments determine a distribution.

Core questions

How is the distribution of a function of random variables computed from the original distribution?
What do the successive moments of a distribution measure?
How do generating functions encode all moments at once?
When do the moments of a distribution determine it uniquely?

Key concepts

change of variables and Jacobian
moments and central moments
variance and covariance
cumulants
moment problem

Key theories

Change-of-variables formula: For a smooth invertible transformation the density of the transformed variable is the original density evaluated at the inverse, scaled by the absolute value of the Jacobian determinant, which is the standard tool for deriving the law of a function of random variables.
Moment and cumulant generating functions: When it exists, the moment generating function encodes all moments through its derivatives at the origin, and its logarithm, the cumulant generating function, has cumulants that add over independent variables, simplifying the study of sums.
The moment problem: Moments determine a distribution uniquely under growth conditions such as Carleman's, but heavy-tailed distributions like the log-normal can share all moments with others, so moments do not always characterize a law.

Clinical relevance

Transformations and moments are everyday tools of applied probability: deriving the distribution of a transformed quantity supports simulation and error propagation, moments give the means, variances, and correlations used throughout statistics and portfolio theory, and skewness and kurtosis flag departures from normality in risk and quality-control analysis.

History

Moments and the moment problem were central to the nineteenth-century work of Chebyshev, Markov, and Stieltjes, who used moment methods to prove early limit theorems; the change-of-variables technique for densities is the probabilistic counterpart of the substitution rule from calculus.

Key figures

Pafnuty Chebyshev
Thomas Stieltjes
William Feller
Carl Friedrich Gauss

Seminal works

feller1971

Frequently asked questions

Do the moments of a distribution always determine it?: Not always; under growth conditions on the moments they do, but some distributions, such as the log-normal, share every moment with distinct distributions, so the moment sequence can fail to pin down the law.
Why introduce cumulants alongside moments?: Cumulants add over independent random variables, so they behave more simply for sums than moments do; the second cumulant is the variance and higher cumulants measure departures from normality, all of which vanish above order two for the normal distribution.