Moment

Compute moment of univariate distributions:

Compute a moment of specific order:

Evaluate a moment of specific order numerically:

Compute moments of multivariate distributions:

Compute moment of a truncated distribution:

Find moment of a user-defined distribution:

Evaluate moment of a function of a random variable:

Compute moments of distributions derived from data:

Compute moment for a set of 5 independent identically distributed samples of size 1000:

TraditionalForm formatting:

Compute a list of univariate moments along the outermost dimension:

Equivalently:

Compute a multivariate moment for the same data:

Find multivariate moment from sparse data:

Compare with a population moment:

Estimate parameters of a distribution using the method of moments:

Compare data and estimated parametric distribution:

Find normal approximation to GammaDistribution using the method of moments:

Show how and depend on and :

Compare original and approximated distribution:

The law of large numbers states that a sample moment approaches a population moment as the sample size increases. Use Histogram to show probability distribution of a second sample moment of uniform random variates for different sample sizes:

Visualize the convergence process:

Build type A Gram-Charlier expansion of order 6:

A monotone PDF with a positive domain is bounded by :

Prove the identity for exponential distribution for the first few orders:

Moments of PearsonDistribution satisfy a three-term recurrence equation implied by the defining differential equation for the density function :

Verify the moment equations:

Use the recurrence equation to express parameters of PearsonDistribution in terms of its moments:

Fit PearsonDistribution to data:

Check that moments of the resulting distribution are equal to moments of data:

Find quadrature rule for approximating the expectation of a function of a random variable:

Find lowest-order orthogonal polynomials:

Check their orthonormality:

Find quadrature points:

Find quadrature weights, requiring rule to be exact on polynomials of order up to :

Compute approximation to expectation of :

Check with NExpectation:

Moment is equivalent to Expectation of a power of a random variable:

A multivariate moment is equivalent to Expectation of a multivariate monomial:

Moment of order one is the Mean for univariate distributions:

Mean of a multivariate distribution is a list of moments of its univariate marginal distributions:

Alternatively, use Moment with orders given by unit vectors:

Moment of order is the same as when both exist:

Use Moment directly:

Find the moment-generating function by using GeneratingFunction:

Compare with direct evaluation of MomentGeneratingFunction:

Moment can be expressed through Cumulant, FactorialMoment, or CentralMoment:

Sample moments are unbiased estimators of population moments:

Hence the sampling distribution expectation of the estimator equals the estimated moment:

Verify this on a sample of fixed size; evaluate the estimator on the sample:

Find its expectation assuming independent identically distributed random variables and :

Heavy-tailed distributions may only have a few moments defined:

Some heavy-tailed distributions have no moments defined:

Often quantiles can be used to characterize distributions:

Find an unbiased estimator for a product of moments:

Two different distributions can have the same sequence of moments:

Compare their densities on log-scale:

Compute their moments:

Prove them equal: