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:

   >>> from sympy import *
       from sympy.stats import *
       init_printing()

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

   >>> X, Y = Die('X', 6), Die('Y', 6)

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

   >>> P(Eq(X, 3))

   

   >>> P(X > 3)

   
4. Conditions can also involve multiple random variables:

   >>> P(X > Y)

   
5. We can compute conditional probabilities:

   >>> P(X + Y > 6, X < 5)

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

   >>> Z = Normal('Z', 0, 1)  # Gaussian...