/

# Statistics¶

The statistics module in SymPy implements standard probability distributions and related tools. Its contents can be imported with the following statement:

>>> from sympy import *
>>> from sympy.statistics import *
>>> import sys
>>> sys.displayhook = pprint


## Normal distributions¶

Normal(mu, sigma) creates a normal distribution with mean value mu and standard deviation sigma. The Normal class defines several useful methods and properties. Various properties can be accessed directly as follows:

>>> N = Normal(0, 1)
>>> N.mean
0
>>> N.median
0
>>> N.variance
1
>>> N.stddev
1


You can generate random numbers from the desired distribution with the random method:

>>> N = Normal(10, 5)
>>> N.random()
4.914375200829805834246144514
>>> N.random()
11.84331557474637897087177407
>>> N.random()
17.22474580071733640806996846
>>> N.random()
9.864643097429464546621602494


The probability density function (pdf) and cumulative distribution function (cdf) of a distribution can be computed, either in symbolic form or for particular values:

>>> N = Normal(1, 1)
>>> x = Symbol('x')
>>> N.pdf(1)
___
\/ 2
--------
____
2*\/ pi
>>> N.pdf(3).evalf()
0.0539909665131881
>>> N.cdf(x)
/  ___        \
|\/ 2 *(x - 1)|
erf|-------------|
\      2      /   1
------------------ + -
2            2
>>> N.cdf(-oo), N.cdf(1), N.cdf(oo)
(0, 1/2, 1)
>>> N.cdf(5).evalf()
0.999968328758167


The method probability gives the total probability on a given interval (a convenient alternative syntax for cdf(b)-cdf(a)):

>>> N = Normal(0, 1)
>>> N.probability(-oo, 0)
1/2
>>> N.probability(-1, 1)
/  ___\
|\/ 2 |
erf|-----|
\  2  /
>>> N.probability(-1, 1).evalf()
0.682689492137086


You can also generate a symmetric confidence interval from a given desired confidence level (given as a fraction 0-1). For the normal distribution, 68%, 95% and 99.7% confidence levels respectively correspond to approximately 1, 2 and 3 standard deviations:

>>> N = Normal(0, 1)
>>> N.confidence(0.68)
(-0.994457883209753, 0.994457883209753)
>>> N.confidence(0.95)
(-1.95996398454005, 1.95996398454005)
>>> N.confidence(0.997)
(-2.96773792534178, 2.96773792534178)


Plug the interval back in to see that the value is correct:

>>> N.probability(*N.confidence(0.95)).evalf()
0.950000000000000


## Other distributions¶

Besides the normal distribution, uniform continuous distributions are also supported. Uniform(a, b) represents the distribution with uniform probability on the interval [a, b] and zero probability everywhere else. The Uniform class supports the same methods as the Normal class.

Additional distributions, including support for arbitrary user-defined distributions, are planned for the future.