Moments of a normal distribution

[Justin Vincent]

Wanted to get the moments of a Normal distribution WRT mu and sigma, a la:

http://en.wikipedia.org/wiki/Normal_distribution#Moments

For starters, this outputs 1 for one for one crazy domains, and then leaves the integral unevaluated on the rest of the domain. But it gets one, so I'm cool with that.

from sympy.statistics import Normal

from sympy import symbols, integrate

f = Normal(mu, sigma).pdf(x)

x, mu, sigma = symbols('x mu sigma')

integrate(f, (x,-oo, oo))

Then when I get the first moment, it really starts to go crazy though, and leaves it all indefinite:

integrate(f * x, (x, -oo, oo))

What approach should I take here if I wanted to get those values cited in wikipedia? Haven't used Sympy much before. Thanks in advance.

Justin

[Matthew Rocklin]

Use sympy.stats, not sympy.statistics

In [1]: x, mu = symbols('x,mu', real=True)

In [2]: sigma = symbols('sigma', positive=True)

In [3]: from sympy.stats import *

In [4]: X = Normal('X', mu, sigma)

In [5]: moment?

Type:        function

String form: <function moment at 0x7fc3fbbe2b18>

File:        /home/mrocklin/workspace/sympy/sympy/stats/rv_interface.py

Definition:  moment(X, n, c=0, condition=None, **kwargs)

Docstring:

Return the nth moment of a random expression about c i.e. E((X-c)**n)

Default value of c is 0.

Examples

========

>>> from sympy.stats import Die, moment, E

>>> X = Die('X', 6)

>>> moment(X, 1, 6)

-5/2

>>> moment(X, 2)

91/6

>>> moment(X, 1) == E(X)

True

In [6]: moment(X, 1)

Out[6]: μ

In [7]: moment(X, 2)

Out[7]: 

 2    2

μ  + σ 

In [8]: moment(X, 3)

Out[8]: 

 3        2

μ  + 3⋅μ⋅σ 

In [9]: moment(X, 4)

Out[9]: 

 4      2  2      4

μ  + 6⋅μ ⋅σ  + 3⋅σ 

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[F. B.]

On Tuesday, July 15, 2014 11:28:11 PM UTC+2, Matthew wrote:

Use sympy.stats, not sympy.statistics

sympy.statistics has been removed from the master branch. It will no longer exist as soon as the next SymPy version is released.

[Justin Vincent]

Thanks! This works great.

[Justin Vincent]

A little notebook that came out of this work, is case anyone is interested:

http://nbviewer.ipython.org/github/justinvf/algorithmshop/blob/master/20140715-mixture-models/pearson_gaussian_mixtures.ipynb

[Matthew Rocklin]

Cool.  It's nice to see SymPy used in this way.

To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/8e7e3a2a-639c-4816-a892-872e94e10ac9%40googlegroups.com.