This page summarizes how to work with univariate probability distributions using Python’s SciPy library. See also notes on working with distributions in Mathematica, Excel, and R/SPLUS.
Probability distribution classes are located in scipy.stats
.
The methods on continuous distribution classes are as follows.
Method  Meaning 

pdf 
Probability density function 
cdf 
Cumulative distribution function 
sf 
Survival function = complementary CDF 
ppf 
Percentile point function (i.e. CDF inverse) 
isf 
Inverse survival function (Complementary CDF inverse) 
stats 
Mean, variance, skew, or kurtosis 
moment 
Noncentral moments 
rvs 
Random samples 
Functions such as pdf
and cdf
are defined over the entire real line. For example, the beta distribution is commonly defined on the interval [0, 1]. If you ask for the pdf
outside this interval, you simply get 0. If you ask for the cdf
to the left of the interval you get 0, and to the right of the interval you get 1.
Distributions have a general form and a “frozen” form. The general form is stateless: you supply the distribution parameters as arguments to every call. The frozen form creates an object with the distribution parameters set. For example, you could evaluate the PDF of a normal(3, 4) distribution at the value 5 by
stats.norm.pdf(5, 3, 4)
or by
mydist = stats.norm(3, 4) mydist.pdf(5)
Note that the argument of the PDF, in this example 5, comes before the distribution parameters. Note also that for discrete distributions, one would call pmf
(probability mass function) rather than the pdf
(probability density function).
Distributions and parameterizations
SciPy makes every continuous distribution into a locationscale family, including some distributions that typically do not have location scale parameters. This unusual approach has its advantages. For example, the question of whether an exponential distribtion is parameterized in terms of its mean or its rate goes away: there is no mean or rate parameter per se, only a scale parameter like every other continuous distribution.
The table below only lists parameters in addition to location and scale.
Distribution  SciPy name  Parameters 

beta  beta 
shape1 , shape2 
binomial  binom 
size , prob 
Cauchy  cauchy 

chisquared  chi2 
df 
exponential  expon 

F  f 
df1 , df2 
gamma  gamma 
shape 
geometric  geom 
p 
hypergeometric  hypergeom 
M , n , N 
inverse gamma  invgamma 
shape 
lognormal  lognorm 
sdlog 
logistic  logistic 

negative binomial  nbinom 
size , prob 
normal  norm 

Poisson  poisson 
lambda 
Student t  t 
df 
uniform  unif 

Weibull  exponweib 
exponent , shape 
SciPy does not have a simple Weibull distribution but instead has a generalization of the Weibull called the exponentiated Weibull. Set the exponential parameter to 1 and you get the ordinary Weibull distribution.
The hypergeometric distribution gives the probability of various numbers of red balls when N balls are taken from an urn containing n red balls and M–n blue balls. Note that another popular convention uses the number of red and blue balls rather than the number of red balls and the total number of balls.
Note that the parameters for the lognormal are the mean and standard deviation of the log of the distribution, not the mean and standard deviation of the distribution itself.
The PDF or PMF of a distribution is contained in the extradoc
string. For example:
>>> stats.poisson.extradoc Poisson distribution poisson.pmf(k, mu) = exp(mu) * mu**k / k! for k >= 0
The lognormal distribution as implemented in SciPy may not be the same as the lognormal distribution implemented elsewhere. When the location parameter is 0, the stats.lognorm
with parameter s
corresponds to a lognormal(0, s) distribution as defined here. But if the location parameter is not 0, stats.lognorm
does not correspond to a lognormal distribution under the other distribution. The difference is whether the PDF contains log(xμ) or log(x) – μ.
For more information, see scipy.stats online documentation.
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