Create an interpolated smooth density estimate for some data:

Compute probabilities from the distribution:

Create an interpolated version of a kernel density estimate for data with quantities:

Find moments:

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:

Compare with SmoothKernelDistribution:

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 0.1 and 1:

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 0 (none) to 1 (full):

Vary the initial bandwidth for an adaptive estimate:

Specify an initial bandwidth of 1.0 and 0.1, 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[0,1]:

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:

Use bounding to stay within the domain:

Define the smooth kernel distribution with a Gaussian kernel:

The support for PDF extends beyond the data support:

Impose bounds:

Compare to the original BetaDistribution:

Compare smooth kernel distributions with a bounded Epanechnikov kernel:

The bounded smooth kernel density estimate is more accurate at the boundaries:

Use a bounded cosine kernel for two-dimensional data:

Compare the estimated density to the population distribution density:

Bounded Gaussian kernel:

Truncating the ordinary smooth kernel distribution yields a different result:

By default, nonuniform interpolation is used to create a smooth estimate:

Specify the initial number of sample points to use:

Use 4 interpolation points:

A larger number of points yields a smoother estimate:

Specify the number of interpolating points to use for bivariate data:

Use 5 and 50 interpolation points in each dimension:

Use different numbers of interpolation points in each dimension:

Specify 3 and 30 points or 30 and 3:

A smooth result does not imply a high-quality estimate:

Using 1000 interpolation points creates a very smooth estimate in this case:

By default, the estimate extends at most 12 bandwidths beyond the data:

Set the maximum number of bandwidths to use:

Use 0 and 12 bandwidths, respectively:

Set a different number for each endpoint:

Specify the number of extra bandwidths to use for multivariate data:

Use 0 and 12 bandwidths, respectively:

Specify the number of extra bandwidths to use in each dimension:

Use 0 and 12 bandwidths or 12 and 0 bandwidths, respectively:

Set a different number for each endpoint in each dimension:

By default, the number of kernels is generally optimal:

Specify the maximum number of kernels to use in the estimate:

Place at most 5 kernels:

A larger number of kernels gives a better estimate of the underlying distribution:

Place a kernel at each data point:

Vary the bandwidth used for the same number of kernels:

Specify the maximum number of kernels to use in each dimension for bivariate data:

Place at most 10 and 100 kernels, respectively:

Set the maximum number of kernels in each dimension:

Specify a maximum of 5 and 50 kernels or 50 and 5:

A smooth estimate will usually be returned by default:

Specify the maximum number of recursive subdivisions to use:

Give the maximum number of recursive subdivisions for bivariate data:

Use at most 2 and 6 subdivisions, respectively:

Set the maximum number of recursive subdivisions in each dimension:

Specify a maximum of 0 and 3 subdivisions or 3 and 0:

By default, estimates are optimized for a balance between speed and quality:

Set PerformanceGoal for speed or quality or use Automatic to balance the two:

More time is spent with PerformanceGoal set to "Quality":

Use with ControlActive to vary PerformanceGoal dynamically: