# Probability Distributions¶

Probability Distributions

This module provides three types of probability distributions:

• RealDistribution: various real-valued probability distributions.
• SphericalDistribution: uniformly distributed points on the surface of an $$n-1$$ sphere in $$n$$ dimensional euclidean space.
• GeneralDiscreteDistribution: user-defined discrete distributions.

AUTHORS:

• Josh Kantor (2007-02): first version
• William Stein (2007-02): rewrite of docs, conventions, etc.
• Carlo Hamalainen (2008-08): full doctest coverage, more documentation, GeneralDiscreteDistribution, misc fixes.
• Kwankyu Lee (2010-05-29): F-distribution support.

REFERENCES:

class sage.gsl.probability_distribution.GeneralDiscreteDistribution

Create a discrete probability distribution.

INPUT:

• P - list of probabilities. The list will automatically be normalised if sum(P) is not equal to 1.
• rng - (optional) random number generator to use. May be one of 'default', 'luxury', or 'taus'.
• seed - (optional) seed to use with the random number generator.

OUTPUT:

• a probability distribution where the probability of selecting x is P[x].

EXAMPLES:

Constructs a GeneralDiscreteDistribution with the probability distribution $$P$$ where $$P(0) = 0.3$$, $$P(1) = 0.4$$, $$P(2) = 0.3$$:

sage: P = [0.3, 0.4, 0.3]
sage: X = GeneralDiscreteDistribution(P)
sage: X.get_random_element()
1

Checking the distribution of samples:

sage: P = [0.3, 0.4, 0.3]
sage: counts = [0] * len(P)
sage: X = GeneralDiscreteDistribution(P)
sage: nr_samples = 10000
sage: for _ in range(nr_samples):
...       counts[X.get_random_element()] += 1
sage: [1.0*x/nr_samples for x in counts]
[0.304200000000000, 0.397300000000000, 0.298500000000000]

The distribution probabilities will automatically be normalised:

sage: P = [0.1, 0.3]
sage: X = GeneralDiscreteDistribution(P, seed = 0)
sage: counts = [0, 0]
sage: for _ in range(10000):
...       counts[X.get_random_element()] += 1
sage: float(counts[1]/counts[0])
3.042037186742118

TESTS:

Make sure that repeated initializations are randomly seeded (trac ticket #9770):

sage: P = [0.001] * 1000
sage: Xs = [GeneralDiscreteDistribution(P).get_random_element() for _ in range(1000)]
sage: len(set(Xs)) > 2^^32
True

The distribution probabilities must be non-negative:

sage: GeneralDiscreteDistribution([0.1, -0.1])
Traceback (most recent call last):
...
ValueError: The distribution probabilities must be non-negative
get_random_element()

Get a random sample from the probability distribution.

EXAMPLE:

sage: P = [0.3, 0.4, 0.3]
sage: X = GeneralDiscreteDistribution(P)
sage: [X.get_random_element() for _ in range(10)]
[1, 0, 1, 1, 0, 1, 1, 1, 1, 1]
sage: isinstance(X.get_random_element(), sage.rings.integer.Integer)
True
reset_distribution()

This method resets the distribution.

EXAMPLE:

sage: T = GeneralDiscreteDistribution([0.1, 0.3, 0.6])
sage: T.set_seed(0)
sage: [T.get_random_element() for _ in range(10)]
[2, 2, 2, 2, 2, 1, 2, 2, 1, 2]
sage: T.reset_distribution()
sage: [T.get_random_element() for _ in range(10)]
[2, 2, 2, 2, 2, 1, 2, 2, 1, 2]
set_random_number_generator(rng='default')

Set the random number generator to be used by gsl.

EXAMPLE:

sage: X = GeneralDiscreteDistribution([0.3, 0.4, 0.3])
sage: X.set_random_number_generator('taus')
set_seed(seed)

Set the seed to be used by the random number generator.

EXAMPLE:

sage: X = GeneralDiscreteDistribution([0.3, 0.4, 0.3])
sage: X.set_seed(1)
sage: X.get_random_element()
1
class sage.gsl.probability_distribution.ProbabilityDistribution

Bases: object

Concrete probability distributions should be derived from this abstract class.

generate_histogram_data(num_samples=1000, bins=50)

Compute a histogram of the probability distribution.

INPUT:

• num_samples - (optional) number of times to sample from the probability distribution
• bins - (optional) number of bins to divide the samples into.

OUTPUT:

• a tuple. The first element of the tuple is a list of length bins, consisting of the normalised histogram of the random samples. The second list is the bins.

EXAMPLE:

sage: from sage.gsl.probability_distribution import GeneralDiscreteDistribution
sage: P = [0.3, 0.4, 0.3]
sage: X = GeneralDiscreteDistribution(P)
sage: h, b = X.generate_histogram_data(bins = 10)
sage: h
[1.6299999999999999,
0.0,
0.0,
0.0,
0.0,
1.9049999999999985,
0.0,
0.0,
0.0,
1.4650000000000003]
sage: b
[0.0, 0.20000000000000001, 0.40000000000000002, 0.60000000000000009, 0.80000000000000004, 1.0, 1.2000000000000002, 1.4000000000000001, 1.6000000000000001, 1.8, 2.0]
generate_histogram_plot(name, num_samples=1000, bins=50)

Save the histogram from generate_histogram_data() to a file.

INPUT:

• name - file to save the histogram plot (as a PNG).
• num_samples - (optional) number of times to sample from the probability distribution
• bins - (optional) number of bins to divide the samples into.

EXAMPLE:

This saves the histogram plot to my_general_distribution_plot.png in the temporary directory SAGE_TMP:

sage: from sage.gsl.probability_distribution import GeneralDiscreteDistribution
sage: import os
sage: P = [0.3, 0.4, 0.3]
sage: X = GeneralDiscreteDistribution(P)
sage: file = os.path.join(SAGE_TMP, "my_general_distribution_plot")
sage: X.generate_histogram_plot(file)
get_random_element()

To be implemented by a derived class:

sage: P = sage.gsl.probability_distribution.ProbabilityDistribution()
sage: P.get_random_element()
Traceback (most recent call last):
...
NotImplementedError: implement in derived class
class sage.gsl.probability_distribution.RealDistribution

The RealDistribution class provides a number of routines for sampling from and analyzing and visualizing probability distributions. For precise definitions of the distributions and their parameters see the gsl reference manuals chapter on random number generators and probability distributions.

EXAMPLES:

Uniform distribution on the interval [a, b]:

sage: a = 0
sage: b = 2
sage: T = RealDistribution('uniform', [a, b])
sage: T.get_random_element()
0.8175557665526867
sage: T.distribution_function(0)
0.5
sage: T.cum_distribution_function(1)
0.5
sage: T.cum_distribution_function_inv(.5)
1.0

The gaussian distribution takes 1 parameter sigma. The standard gaussian distribution has sigma = 1:

sage: sigma = 1
sage: T = RealDistribution('gaussian', sigma)
sage: T.get_random_element()
-0.5860943109756299
sage: T.distribution_function(0)
0.3989422804014327
sage: T.cum_distribution_function(1)
0.8413447460685429
sage: T.cum_distribution_function_inv(.5)
0.0

The rayleigh distribution has 1 parameter sigma:

sage: sigma = 3
sage: T = RealDistribution('rayleigh', sigma)
sage: T.get_random_element()
5.748307572643492
sage: T.distribution_function(0)
0.0
sage: T.cum_distribution_function(1)
0.054040531093234534
sage: T.cum_distribution_function_inv(.5)
3.532230067546424

The lognormal distribution has two parameters sigma and zeta:

sage: zeta = 0
sage: sigma = 1
sage: T = RealDistribution('lognormal', [zeta, sigma])
sage: T.get_random_element()
0.3876433713532701
sage: T.distribution_function(0)
0.0
sage: T.cum_distribution_function(1)
0.5
sage: T.cum_distribution_function_inv(.5)
1.0

The pareto distribution has two parameters a, and b:

sage: a = 1
sage: b = 1
sage: T = RealDistribution('pareto', [a, b])
sage: T.get_random_element()
10.418714048916407
sage: T.distribution_function(0)
0.0
sage: T.cum_distribution_function(1)
0.0
sage: T.cum_distribution_function_inv(.5)
2.0

The t-distribution has one parameter nu:

sage: nu = 1
sage: T = RealDistribution('t', nu)
sage: T.get_random_element()  # rel tol 1e-15
-8.404911172800615
sage: T.distribution_function(0)      # rel tol 1e-15
0.3183098861837906
sage: T.cum_distribution_function(1)  # rel tol 1e-15
0.75
sage: T.cum_distribution_function_inv(.5)
0.0

The F-distribution has two parameters nu1 and nu2:

sage: nu1 = 9; nu2 = 17
sage: F = RealDistribution('F', [nu1,nu2])
sage: F.get_random_element()  # rel tol 1e-14
1.239233786115256
sage: F.distribution_function(1)  # rel tol 1e-14
0.6695025505192798
sage: F.cum_distribution_function(3.68)  # rel tol 1e-14
0.9899717772300652
sage: F.cum_distribution_function_inv(0.99)  # rel tol 1e-14
3.682241524045864

The chi-squared distribution has one parameter nu:

sage: nu = 1
sage: T = RealDistribution('chisquared', nu)
sage: T.get_random_element()
0.4603367753992381
sage: T.distribution_function(0)
+infinity
sage: T.cum_distribution_function(1)  # rel tol 1e-14
0.6826894921370856
sage: T.cum_distribution_function_inv(.5)  # rel tol 1e-14
0.45493642311957305

The exponential power distribution has two parameters a and b:

sage: a = 1
sage: b = 2.5
sage: T = RealDistribution('exppow', [a, b])
sage: T.get_random_element()
0.16442075306686463
sage: T.distribution_function(0)  # rel tol 1e-14
0.5635302489930136
sage: T.cum_distribution_function(1)  # rel tol 1e-14
0.940263052542855

The beta distribution has two parameters a and b:

sage: a = 2
sage: b = 2
sage: T = RealDistribution('beta', [a, b])
sage: T.get_random_element()  # rel tol 1e-14
0.7110581877139808
sage: T.distribution_function(0)
0.0
sage: T.cum_distribution_function(1)
1.0

The weibull distribution has two parameters a and b:

sage: a = 1
sage: b = 1
sage: T = RealDistribution('weibull', [a, b])
sage: T.get_random_element()
1.1867854542468694
sage: T.distribution_function(0)
1.0
sage: T.cum_distribution_function(1)
0.6321205588285577
sage: T.cum_distribution_function_inv(.5)
0.6931471805599453

It is possible to select which random number generator drives the sampling as well as the seed. The default is the Mersenne twister. Also available are the RANDLXS algorithm and the Tausworthe generator (see the gsl reference manual for more details). These are all supposed to be simulation quality generators. For RANDLXS use rng = 'luxury' and for tausworth use rng = 'taus':

sage: T = RealDistribution('gaussian', 1, rng = 'luxury', seed = 10)

To change the seed at a later time use set_seed:

sage: T.set_seed(100)

TESTS:

Make sure that repeated initializations are randomly seeded (trac ticket #9770):

sage: Xs = [RealDistribution('gaussian', 1).get_random_element() for _ in range(1000)]
sage: len(set(Xs)) > 2^^32
True
cum_distribution_function(x)

Evaluate the cumulative distribution function of the probability distribution at x.

EXAMPLE:

sage: T = RealDistribution('uniform', [0, 2])
sage: T.cum_distribution_function(1)
0.5
cum_distribution_function_inv(x)

Evaluate the inverse of the cumulative distribution distribution function of the probability distribution at x.

EXAMPLE:

sage: T = RealDistribution('uniform', [0, 2])
sage: T.cum_distribution_function_inv(.5)
1.0
distribution_function(x)

Evaluate the distribution function of the probability distribution at x.

EXAMPLES:

sage: T = RealDistribution('uniform', [0, 2])
sage: T.distribution_function(0)
0.5
sage: T.distribution_function(1)
0.5
sage: T.distribution_function(1.5)
0.5
sage: T.distribution_function(2)
0.0
get_random_element()

Get a random sample from the probability distribution.

EXAMPLE:

sage: T = RealDistribution('gaussian', 1, seed = 0)
sage: T.get_random_element()  # rel tol 4e-16
0.13391860811867587
plot(*args, **kwds)

Plot the distribution function for the probability distribution. Parameters to sage.plot.plot.plot.plot can be passed through *args and **kwds.

EXAMPLE:

sage: T = RealDistribution('uniform', [0, 2])
sage: P = T.plot()
reset_distribution()

This method resets the distribution.

EXAMPLE:

sage: T = RealDistribution('gaussian', 1, seed = 10)
sage: [T.get_random_element() for _ in range(10)]  # rel tol 4e-16
[-0.7460999595745819, -0.004644606626413462, -0.8720538317207641, 0.6916259921666037, 2.67668674666043, 0.6325002813661014, -0.7974263521959355, -0.5284976893366636, 1.1353119849528792, 0.9912505673230749]
sage: T.reset_distribution()
sage: [T.get_random_element() for _ in range(10)]  # rel tol 4e-16
[-0.7460999595745819, -0.004644606626413462, -0.8720538317207641, 0.6916259921666037, 2.67668674666043, 0.6325002813661014, -0.7974263521959355, -0.5284976893366636, 1.1353119849528792, 0.9912505673230749]
set_distribution(name='uniform', parameters=[])

This method can be called to change the current probability distribution.

EXAMPLES:

sage: T = RealDistribution('gaussian', 1)
sage: T.set_distribution('gaussian', 1)
sage: T.set_distribution('pareto', [0, 1])
set_random_number_generator(rng='default')

Set the gsl random number generator to be one of default, luxury, or taus.

EXAMPLE:

sage: T = SphericalDistribution()
sage: T.set_random_number_generator('default')
sage: T.set_seed(0)
sage: T.get_random_element()  # rel tol 4e-16
(0.07961564104639995, -0.05237671627581255, 0.9954486572862178)
sage: T.set_random_number_generator('luxury')
sage: T.set_seed(0)
sage: T.get_random_element()  # rel tol 4e-16
(0.07961564104639995, -0.05237671627581255, 0.9954486572862178)
set_seed(seed)

Set the seed for the underlying random number generator.

EXAMPLE:

sage: T = RealDistribution('gaussian', 1, rng = 'luxury', seed = 10)
sage: T.set_seed(100)
class sage.gsl.probability_distribution.SphericalDistribution

This class is capable of producing random points uniformly distributed on the surface of an n-1 sphere in n dimensional euclidean space. The dimension, n is selected via the keyword dimension. The random number generator which drives it can be selected using the keyword rng. Valid choices are default which uses the Mersenne-Twister, luxury which uses RANDLXS, and taus which uses the tausworth generator. The default dimension is 3.

EXAMPLES:

sage: T = SphericalDistribution()
sage: T.get_random_element()  # rel tol 1e-14
(-0.2922296724828204, -0.9563459345927822, 0.0020668595602153454)
sage: T = SphericalDistribution(dimension = 4, rng = 'luxury')
sage: T.get_random_element()  # rel tol 1e-14
(-0.0363300434761631, 0.6459885817544098, 0.24825817345598158, 0.7209346430129753)

TESTS:

Make sure that repeated initializations are randomly seeded (trac ticket #9770):

sage: Xs = [tuple(SphericalDistribution(2).get_random_element()) for _ in range(1000)]
sage: len(set(Xs)) > 2^^32
True
get_random_element()

Get a random sample from the probability distribution.

EXAMPLE:

sage: T = SphericalDistribution(seed = 0)
sage: T.get_random_element()  # rel tol 4e-16
(0.07961564104639995, -0.05237671627581255, 0.9954486572862178)
reset_distribution()

This method resets the distribution.

EXAMPLE:

sage: T = SphericalDistribution(seed = 0)
sage: [T.get_random_element() for _ in range(4)]  # rel tol 4e-16
[(0.07961564104639995, -0.05237671627581255, 0.9954486572862178), (0.4123599490593727, 0.5606817859360097, -0.7180495855658982), (-0.9619860891623148, -0.2726473494040498, -0.015690351211529927), (0.5674297579435619, -0.011206783800420301, -0.8233455397322326)]
sage: T.reset_distribution()
sage: [T.get_random_element() for _ in range(4)]  # rel tol 4e-16
[(0.07961564104639995, -0.05237671627581255, 0.9954486572862178), (0.4123599490593727, 0.5606817859360097, -0.7180495855658982), (-0.9619860891623148, -0.2726473494040498, -0.015690351211529927), (0.5674297579435619, -0.011206783800420301, -0.8233455397322326)]
set_random_number_generator(rng='default')

Set the gsl random number generator to be one of default, luxury, or taus.

EXAMPLE:

sage: T = SphericalDistribution()
sage: T.set_random_number_generator('default')
sage: T.set_seed(0)
sage: T.get_random_element()  # rel tol 4e-16
(0.07961564104639995, -0.05237671627581255, 0.9954486572862178)
sage: T.set_random_number_generator('luxury')
sage: T.set_seed(0)
sage: T.get_random_element()  # rel tol 4e-16
(0.07961564104639995, -0.05237671627581255, 0.9954486572862178)
set_seed(seed)

Set the seed for the underlying random number generator.

EXAMPLE:

sage: T = SphericalDistribution(seed = 0)
sage: T.set_seed(100)

Probability

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Random variables and probability spaces