Create an interpolated smooth density estimate for some data:

Compute probabilities from the distribution:

Increase the bandwidth for smoother estimates:

Allow the bandwidth to vary adaptively with local density:

Interpolate kernel density estimates in higher dimensions:

Plot the univariate marginal PDFs:

Plot the bivariate marginal PDFs:

Select from built-in kernel functions or build a custom one:

A custom kernel function:

Specify radial or product type kernels for multivariate estimates:

Estimate distribution functions:

Compute moments of the distribution:

Special moments:

General moments:

Quantile function:

Special quantile values:

Generate random numbers:

Compute probabilities and expectations:

Estimate bivariate distribution functions:

Compute moments of a bivariate distribution:

Special moments:

General moments:

Generate random numbers:

Show the point distribution:

Automatically select the bandwidth to use:

More data yields better approximations to the underlying distribution:

Explicitly specify the bandwidth to use:

Use bandwidths of

and

:

Larger bandwidths yield smoother estimates:

Specify bandwidths in units of standard deviation:

Use bandwidths of

and

the standard deviation:

Allow the bandwidth to vary adaptively with local density:

Vary the local sensitivity from

(none) to

(full):

Vary the initial bandwidth for an adaptive estimate:

Specify an initial bandwidth of

and

, respectively:

Use any of several automatic bandwidth selection methods:

Silverman's method is used by default:

The PDFs are equivalent:

By default, Silverman's method is used to independently select bandwidths in each dimension:

Any automated method can be used to independently select diagonal bandwidth elements:

Methods used to estimate the bandwidth diagonal need not be the same:

Use adaptive, oversmoothed, and constant bandwidths in the respective dimensions:

Plot the univariate marginal PDFs:

Give a scalar value to use the same bandwidth in all dimensions:

To use nonzero off-diagonal elements, give a fully specified bandwidth matrix:

Specify any one of several kernel functions:

Define the kernel function as a pure function:

By default, the Gaussian kernel is used:

This is equivalent to using the PDF of a

NormalDistribution:

Shapes of some univariate kernel functions:

Specify any one of several kernel functions for multivariate data:

Choose between product and radial-type kernel functions for multivariate data: