stats.tex

%% Stats
The \verb|sympy.stats| submodule provides random variable types and methods for
computing statistical properties of expressions involving random
variables, which can be either continuous or discrete, the latter ones being
further divided into finite and infinite. The variables are associated
with probability densities on corresponding domains and internally defined
in terms of probability spaces.
Apart from the possibility of defining the random variables from a user supplied
density distribution, SymPy provides definitions of most common
distributions, including \texttt{Uniform}, \texttt{Poisson}, \texttt{Normal},
\texttt{Binomial}, \texttt{Bernoulli}, and many others.

Properties of random expressions can be calculated using, e.g.,
\texttt{expectation} (abbreviated \texttt{E}) and \texttt{variance} to
calculate expectation and variance. Internally, these functions generate
integrals and summations, which are automatically evaluated. The evaluation
can be suppressed using \texttt{evaluate=False} keyword argument.

Conditions on random variables can be defined with inequalities, equalities,
and logical operators and their overall probabilities are obtained using
\texttt{P}. The features can be illustrated on a model of two dice throws:
\begin{verbatim}
>>> from sympy.stats import Die, P, E
>>> X, Y = Die("X"), Die("Y")
>>> P(Eq(X, 6) & Eq(Y, 6))
1/36
>>> P(X>Y)
5/12
\end{verbatim}
The conditions can also be supplied as a second parameter to \texttt{E},
\texttt{P}, and other methods to calculate the property given the condition:
\begin{verbatim}
>>> E(X, X+Y<5)
5/3
\end{verbatim}

Using the facilities of the \texttt{sympy.stats} submodule, one can, for
example, calculate
%Using the above-mentioned features, it is easy to calculate, for example
the well known properties of the Maxwellian velocity distribution.
\begin{verbatim}
>>> from sympy.stats import Maxwell, density
>>> kT, m, t = symbols("kT m t", positive=True)
>>> v = Maxwell("v", sqrt(kT/m))
>>> E(v)                                # mean velocity
2*sqrt(2)*sqrt(kT)/(sqrt(pi)*sqrt(m))
>>> E(v, evaluate=False)                # unevaluated mean velocity
Integral(sqrt(2)*m**(3/2)*v**3*exp(-m*v**2/(2*kT))/(sqrt(pi)*kT**(3/2)),
(v, 0, oo))
>>> E(m*v**2/2)                         # mean energy
3*kT/2
>>> solve(density(v)(t).diff(t), t)[0]  # most probable velocity
sqrt(2)*sqrt(kT)/sqrt(m)
\end{verbatim}

More information on the \texttt{sympy.stats} submodule can be found in
\cite{rocklin2012symbolic}.