Generating Functions for sympy.stats

[mrocklin]

Is there any benefit to reasoning about generating functions (moment, cumulant, etc...) for random variables? Are there problems that we could solve more easily with this extra formalism? http://en.wikipedia.org/wiki/Cumulant_generating_function http://en.wikipedia.org/wiki/Moment_generating_function

Original issue for #6323: http://code.google.com/p/sympy/issues/detail?id=3224
Original author: https://code.google.com/u/109882876523836932473/

[mrocklin]

http://en.wikipedia.org/wiki/Probability-generating_function http://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c1
Original author: https://code.google.com/u/109882876523836932473/

[aaaagrawal]

@mrocklin : Now that we have functions to calculate moments after https://github.com/sympy/sympy/pull/1875 , do we need these functions ? If yes I would like to work on them.

Ankit.

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c2
Original author: https://code.google.com/u/110118527174861881051/

[mrocklin]

Do we need these?  No.
Would they be useful?  Maybe, I don't know enough about them.
Would they be nice to have in any event?  Sure.

If you want to work on them then that's great.  If nothing else it'd be nice to have their formal definition in SymPy.  As you're working on them it might be interesting to think about problems that these bits of formalism solve that our current system finds difficult.

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c3
Original author: https://code.google.com/u/109882876523836932473/

[aaaagrawal]

Sure, I will think of the cases where these generating/characteristics
functions can be used, as it would serve as better examples in docs too.
Also I have some submission deadlines so I will commit tomorrow. Thanks.

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c4
Original author: https://code.google.com/u/110118527174861881051/

[asmeurer]

We have moved issues to GitHub https://github.com/sympy/sympy/issues .

**Labels:** Restrict-AddIssueComment-Commit  

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c5
Original author: https://code.google.com/u/asmeurer@gmail.com/

[asmeurer]

We have moved issues to GitHub https://github.com/sympy/sympy/issues .

Original comment: http://code.google.com/p/sympy/issues/detail?id=3224#c6
Original author: https://code.google.com/u/asmeurer@gmail.com/

[Smit-create]

Currently sympy.stats support, moment, cmoment, smoment, factorial_moment, characteristic_function, and moment_generating_function

sympy/sympy/stats/rv_interface.py

Lines 19 to 22 in 3b4c1a6

	def moment(X, n, c=0, condition=None, **kwargs):
	"""
	Return the nth moment of a random expression about c i.e. E((X-c)**n)
	Default value of c is 0.

sympy/sympy/stats/rv_interface.py

Lines 197 to 201 in 3b4c1a6

	def cmoment(X, n, condition=None, **kwargs):
	"""
	Return the nth central moment of a random expression about its mean
	i.e. E((X - E(X))**n)

sympy/sympy/stats/rv_interface.py

Lines 218 to 222 in 3b4c1a6

	def smoment(X, n, condition=None, **kwargs):
	"""
	Return the nth Standardized moment of a random expression i.e.
	E(((X - mu)/sigma(X))**n)

sympy/sympy/stats/rv_interface.py

Lines 314 to 319 in 3b4c1a6

	def factorial_moment(X, n, condition=None, **kwargs):
	"""
	The factorial moment is a mathematical quantity defined as the expectation
	or average of the falling factorial of a random variable.
	factorial_moment(X, n) = E(X*(X - 1)*(X - 2)*...*(X - n + 1))

sympy/sympy/stats/rv.py

Lines 969 to 973 in 3b4c1a6

	def characteristic_function(expr, condition=None, evaluate=True, **kwargs):
	"""
	Characteristic function of a random expression, optionally given a second condition
	Returns a Lambda

sympy/sympy/stats/rv.py

Lines 1002 to 1006 in 3b4c1a6

	def moment_generating_function(expr, condition=None, evaluate=True, **kwargs):
	if condition is not None:
	return moment_generating_function(given(expr, condition, **kwargs), **kwargs)
	result = pspace(expr).compute_moment_generating_function(expr, **kwargs)

IMHO, this can be safely closed. Should we close this @Upabjojr @czgdp1807 ?

[Upabjojr]

Currently sympy.stats support, moment, cmoment, smoment, factorial_moment, characteristic_function, and moment_generating_function

Do we have the symbolic class for all of them? It's important to have a class to express unevaluated functions, so we can leave the unevaluated Moment(X, 1) in case functions like moment(X, 1) are unable to evaluate.

[Smit-create]

For unevaluated results, we can use evaluate=False:

In [1]: from sympy.stats import *

In [2]: X = Normal('X', 1, 3)

In [3]: moment(X, 4, evaluate=False)
Out[3]:
∞
⌠
⎮                 2

⎮         -(X - 1)
⎮         ──────────
⎮      4      18
⎮  √2⋅X ⋅ℯ
⎮  ───────────────── dX
⎮         6⋅√π
⌡
-∞

In [4]: moment(X, 4)
Out[4]: 298

In [5]: cmoment(X, 3, evaluate=False)
Out[5]:
∞
⌠
⎮                                 3
⎮     ⎛    ∞                     ⎞
⎮     ⎜    ⌠                     ⎟
⎮     ⎜    ⎮                2    ⎟
⎮     ⎜    ⎮        -(X - 1)     ⎟           2
⎮     ⎜    ⎮        ──────────   ⎟   -(X - 1)
⎮     ⎜    ⎮            18       ⎟   ──────────
⎮     ⎜    ⎮  √2⋅X⋅ℯ             ⎟       18
⎮  √2⋅⎜X - ⎮  ──────────────── dX⎟ ⋅ℯ

⎮     ⎜    ⎮        6⋅√π         ⎟
⎮     ⎜    ⌡                     ⎟
⎮     ⎝    -∞                    ⎠
⎮  ──────────────────────────────────────────── dX
⎮                      6⋅√π
⌡
-∞

In [6]: cmoment(X, 3)
Out[6]: 0

In [7]: factorial_moment(X, 2)
Out[7]: 9

In [8]: factorial_moment(X, 2, evaluate=False)
Out[8]:

∞
⌠
⎮                        2
⎮                -(X - 1)
⎮                ──────────
⎮                    18
⎮  √2⋅X⋅(X - 1)⋅ℯ
⎮  ──────────────────────── dX
⎮            6⋅√π
⌡
-∞

[oscarbenjamin]

Is there a Moment class rather than Integral?

[Upabjojr]

For unevaluated results, we can use evaluate=False

That is returning the unevaluated integral, that is not the unevaluated Moment. We have already classes Probability, Expectation, Variance, Covariance for unevaluated expressions... but not for the other ones.

[Smit-create]

Sure I get it, It Will be nice additions.