Computing exact probabilities and manipulating random variables

SymPy includes a module named stats that lets us create and manipulate random variables. This is useful when we work with probabilistic or statistical models; we can compute symbolic expectancies, variances probabilities, and densities of random variables.

How to do it...

1. Let's import SymPy and the stats module:

   In [1]: from sympy import *
           from sympy.stats import *
           init_printing()

2. Let's roll two dice, X and Y, with six faces each:

   In [2]: X, Y = Die('X', 6), Die('Y', 6)

3. We can compute probabilities defined by equalities (with the Eq operator) or inequalities:

   In [3]: P(Eq(X, 3))
   Out[3]: 1/6
   In [4]: P(X>3)
   Out[4]: 1/2

4. Conditions can also involve multiple random variables:

   In [5]: P(X>Y)
   Out[5]: 5/12

5. We can compute conditional probabilities:

   In [6]: P(X+Y>6, X<5)
   Out[6]: 5/12

6. We can also work with arbitrary discrete or continuous random variables:

   In [7]: Z = Normal('Z',...