Exact input yields exact output:

Approximate input yields approximate output:

Find moments of WeightedData:

Find a moment of EventData:

Find a moment of TimeSeries:

The moment depends only on the values:

Find a moment for data involving quantities:

For a matrix, Moment gives columnwise moments:

For an array, Moment gives columnwise moments at the first level:

Multivariate Moment for an array:

Works with large arrays:

When the input is an Association, Moment works on its values:

SparseArray data can be used just like dense arrays:

Find the moment of a QuantityArray:

Channelwise moment of an RGB image:

Moment intensity value of a grayscale image:

On audio objects, Moment works channelwise:

Scalar moment for univariate distributions:

Scalar moment for multivariate distributions:

Joint moment for multivariate distributions:

Compute a moment for a symbolic order r:

A moment may only evaluate for specific orders:

A moment may only evaluate numerically:

Moments for derived distributions:

Data distribution:

Moment function for a random process:

Find a moment of TemporalData at a time t=0.5:

Find the corresponding moment function together with all the simulations:

TraditionalForm formatting for formal moments:

Convert combinations of formal moments to an expression involving Moment:

Evaluate an expression involving formal moments μ₂+μ₃ for a distribution:

Evaluate for data:

Find a sample estimator for an expression involving Moment:

Evaluate the resulting estimator for data:

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:

Compute a moving moment of a time series data:

Use the window of length .1:

Compute moments for slices of a collection of paths of a random process:

Choose a few slice times:

Plot the fourth moments over these paths:

Estimate parameters of a distribution using the method of moments:

Compare the data with the estimated parametric distribution:

Find normal approximation to GammaDistribution using the method of moments:

Show how and depend on and :

Compare the original and the approximated distributions:

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:

Two different distributions can have the same sequence of moments:

Compare their densities on log-scale:

Compute their moments:

Prove them equal for all non-negative integer orders:

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:

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: