# Continuous Distribution Measures (Moments & Cumulants)¶

This module provides functions to compute moments and cumulants from continuous probability distribution functions.

moment_generating_function(generating_function, symbols, symbols_in_result, velocity=None)

Computes the moment generating function of a probability distribution. It is defined as:

$F[f(\mathbf{x})](\mathbf{t}) = \int e^{<\mathbf{x}, \mathbf{t}>} f(x)\; dx$
Parameters
• generating_function – sympy expression

• symbols – a sequence of symbols forming the vector x

• symbols_in_result – a sequence forming the vector t

• velocity – if the generating function generates central moments, the velocity needs to be substracted. Thus the velocity symbols need to be passed. All generating functions need to have the same parameters.

Returns

an expression that depends now on symbols_in_result (symbols have been integrated out)

Return type

transformation result F

Note

This function uses sympys symbolic integration mechanism, which may not work or take a large amount of time for some functions. Therefore this routine does some transformations/simplifications on the function first, which are taylored to expressions of the form exp(polynomial) i.e. Maxwellian distributions, so that these kinds of functions can be integrated quickly.

central_moment_generating_function(func, symbols, symbols_in_result, velocity=(u_0, u_1, u_2))

Computes central moment generating func, which is defined as:

$K( \vec{\Xi} ) = \exp ( - \vec{\Xi} \cdot \vec{u} ) M( \vec{\Xi}.$

For parameter description see moment_generating_function().

cumulant_generating_function(func, symbols, symbols_in_result, velocity=None)

Computes cumulant generating func, which is the logarithm of the moment generating func:

$C(\vec{\Xi}) = \log M(\vec{\Xi})$

For parameter description see moment_generating_function().

continuous_moment(func, moment, symbols=None)

Computes moment of given function.

Parameters
• func – function to compute moments of

• moment – tuple or polynomial describing the moment

• symbols – if moment is given as polynomial, pass the moment symbols, i.e. the dof of the polynomial

continuous_central_moment(func, moment, symbols=None, velocity=(u_0, u_1, u_2))

Computes central moment of given function.

Parameters
• func – function to compute moments of

• moment – tuple or polynomial describing the moment

• symbols – if moment is given as polynomial, pass the moment symbols, i.e. the dof of the polynomial

continuous_cumulant(func, moment, symbols=None)

Computes cumulant of continuous function.

for parameter description see continuous_moment()