# DistributionCategory R¶

Category of distributions formally given by moments.

0: %

`0` is the Dirac distribution

=: (%, %) -> Boolean

from BasicType

^: (%, PositiveInteger) -> %

`x^k` constructs the distribution of the `k`th power of the random variable with distribution `X` by picking every `k`-th moment.

~=: (%, %) -> Boolean

from BasicType

booleanConvolution: (%, %) -> %

`booleanConvolution(x, y)` returns the boolean convolution of the distributions `x` and `y`

booleanCumulant: (%, PositiveInteger) -> R

`booleanCumulant(x, n)` returns the `n`-th boolean cumulant of the distribution `x`

booleanCumulants: % -> Sequence R

`booleanCumulants(x)` returns the sequence of boolean cumulants of the distribution `x`.

classicalConvolution: (%, %) -> %

`classicalConvolution(x, y)` returns the classical convolution of the distributions `x` and `y`

classicalCumulant: (%, PositiveInteger) -> R

`classicalCumulant(x, n)` returns the `n`-th classical cumulant of the distribution `x`

classicalCumulants: % -> Sequence R

`classicalCumulants(x)` returns sequence of classical cumulants of the distribution `x`

coerce: % -> OutputForm
freeConvolution: (%, %) -> %

`freeConvolution(x, y)` returns the free convolution of the distributions `x` and `y`

freeCumulant: (%, PositiveInteger) -> R

`freeCumulant(x, n)` returns the `n`-th free cumulant of the distribution `x`

freeCumulants: % -> Sequence R

`freeCumulants(x)` returns the sequence of free cumulants of the distribution `x`.

hankelDeterminants: % -> Stream R

`hankelDeterminants(x)` returns the stream of hankel determinants of the distribution `x`.

jacobiParameters: % -> Record(an: Stream Fraction R, bn: Stream Fraction R) if R hasn’t Field and R has IntegralDomain

`jacobiParameters(x)` returns the pair of streams of Jacobi parameters of the distribution `x`.

jacobiParameters: % -> Record(an: Stream R, bn: Stream R) if R has Field

`jacobiParameters(x)` returns the pair of streams of Jacobi parameters of the distribution `x`.

latex: % -> String

from SetCategory

moment: (%, NonNegativeInteger) -> R

`moment(x, n)` returns the `n`-th moment of the distribution `x`

moments: % -> Sequence R

`moments(x)` returns the sequence of moments of the distribution `x`

monotoneConvolution: (%, %) -> %

`monotoneConvolution(x, y)` returns the monotone convolution of the distributions `x` and `y`

monotoneCumulants: % -> Sequence R if R has Algebra Fraction Integer

`monotoneCumulants(x)` returns the sequence of monotone cumulants of the distribution `x`.

orthogonalConvolution: (%, %) -> %

`orthogonalConvolution(x, y)` returns the orthogonal convolution of the distributions `x` and `y`

orthogonalPolynomials: % -> Stream SparseUnivariatePolynomial Fraction R if R hasn’t Field and R has IntegralDomain

`orthogonalPolynomials(x)` returns the stream of orthogonal polynomials.

orthogonalPolynomials: % -> Stream SparseUnivariatePolynomial R if R has Field

`orthogonalPolynomials(x)` returns the stream of orthogonal polynomials.

subordinationConvolution: (%, %) -> %

`subordinationConvolution(x, y)` returns the subordination convolution of the distributions `x` and `y`

BasicType

SetCategory