# 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

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