#!/usr/bin/env python
# coding: utf-8

# > This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.
# 

# # 15.4. Computing exact probabilities and manipulating random variables

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


# ## Rolling dice

# Let's roll two dices X and Y.

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


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

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P(Eq(X, 3))


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P(X>3)


# Conditions can also involve multiple random variables...

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P(X>Y)


# Conditional probabilities...

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


# ## Continuous random variables

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

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Z = Normal('Z', 0, 1)  # Gaussian variable


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P(Z>pi)


# We can compute expectancies and variances...

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E(Z**2), variance(Z**2)


# as well as densities.

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f = density(Z)


# This is a lambda function, it can be evaluated on a SymPy symbol:

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var('x')
f(x)


# We can plot this density.

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get_ipython().run_line_magic('matplotlib', 'inline')
plot(f(x), (x, -6, 6));


# SymPy.stats works by using integrals and summations for computing probabilistic quantities. For example, P(Z>pi) is:

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Eq(Integral(f(x), (x, pi, oo)), 
   simplify(integrate(f(x), (x, pi, oo))))


# > You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).
# 
# > [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages).