Statistics Module

Statistic Module

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

  from sympy.statistics import *

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:

  >>> from sympy.statistics import *
  >>> 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()                  #doctest: +SKIP
  4.914375200829805834246144514
  >>> N.random()                  #doctest: +SKIP
  11.84331557474637897087177407
  >>> N.random()                  #doctest: +SKIP
  17.22474580071733640806996846
  >>> N.random()                  #doctest: +SKIP
  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:

  >>> from sympy import Symbol, oo
  >>> from sympy.statistics import *
  >>> N = Normal(1, 1)
  >>> x = Symbol('x')
  >>> N.pdf(1)
  2**(1/2)/(2*pi**(1/2))
  >>> N.pdf(3).evalf()    #doctest: +SKIP
  0.05399096651318805195056420043
  >>> N.cdf(x)
  erf(2**(1/2)*(x - 1)/2)/2 + 1/2
  >>> 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)
  (1/2)*erf(2*(1/2)**(1/2))
  >>> _.evalf()
  0.477249868051821

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)
  (-2.96773792534179, 2.96773792534179)

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

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

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. Category:Modules