# Stats¶

SymPy statistics module

Introduces a random variable type into the SymPy language.

Random variables may be declared using prebuilt functions such as Normal, Exponential, Coin, Die, etc… or built with functions like FiniteRV.

Queries on random expressions can be made using the functions

 Expression Meaning P(condition) Probability E(expression) Expected value H(expression) Entropy variance(expression) Variance density(expression) Probability Density Function sample(expression) Produce a realization where(condition) Where the condition is true

## Examples¶

>>> from sympy.stats import P, E, variance, Die, Normal
>>> from sympy import Eq, simplify
>>> X, Y = Die('X', 6), Die('Y', 6) # Define two six sided dice
>>> Z = Normal('Z', 0, 1) # Declare a Normal random variable with mean 0, std 1
>>> P(X>3) # Probability X is greater than 3
1/2
>>> E(X+Y) # Expectation of the sum of two dice
7
>>> variance(X+Y) # Variance of the sum of two dice
35/6
>>> simplify(P(Z>1)) # Probability of Z being greater than 1
1/2 - erf(sqrt(2)/2)/2


## Random Variable Types¶

### Finite Types¶

sympy.stats.DiscreteUniform(name, items)[source]

Create a Finite Random Variable representing a uniform distribution over the input set.

Returns a RandomSymbol.

Examples

>>> from sympy.stats import DiscreteUniform, density
>>> from sympy import symbols

>>> X = DiscreteUniform('X', symbols('a b c')) # equally likely over a, b, c
>>> density(X).dict
{a: 1/3, b: 1/3, c: 1/3}

>>> Y = DiscreteUniform('Y', list(range(5))) # distribution over a range
>>> density(Y).dict
{0: 1/5, 1: 1/5, 2: 1/5, 3: 1/5, 4: 1/5}


References

R698

https://en.wikipedia.org/wiki/Discrete_uniform_distribution

R699

http://mathworld.wolfram.com/DiscreteUniformDistribution.html

sympy.stats.Die(name, sides=6)[source]

Create a Finite Random Variable representing a fair die.

Returns a RandomSymbol.

Examples

>>> from sympy.stats import Die, density
>>> from sympy import Symbol

>>> D6 = Die('D6', 6) # Six sided Die
>>> density(D6).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}

>>> D4 = Die('D4', 4) # Four sided Die
>>> density(D4).dict
{1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4}

>>> n = Symbol('n', positive=True, integer=True)
>>> Dn = Die('Dn', n) # n sided Die
>>> density(Dn).dict
Density(DieDistribution(n))
>>> density(Dn).dict.subs(n, 4).doit()
{1: 1/4, 2: 1/4, 3: 1/4, 4: 1/4}

sympy.stats.Bernoulli(name, p, succ=1, fail=0)[source]

Create a Finite Random Variable representing a Bernoulli process.

Returns a RandomSymbol

Examples

>>> from sympy.stats import Bernoulli, density
>>> from sympy import S

>>> X = Bernoulli('X', S(3)/4) # 1-0 Bernoulli variable, probability = 3/4
>>> density(X).dict
{0: 1/4, 1: 3/4}

>>> X = Bernoulli('X', S.Half, 'Heads', 'Tails') # A fair coin toss
>>> density(X).dict


References

R700

https://en.wikipedia.org/wiki/Bernoulli_distribution

R701

http://mathworld.wolfram.com/BernoulliDistribution.html

sympy.stats.Coin(name, p=1/2)[source]

Create a Finite Random Variable representing a Coin toss.

Probability p is the chance of gettings “Heads.” Half by default

Returns a RandomSymbol.

Examples

>>> from sympy.stats import Coin, density
>>> from sympy import Rational

>>> C = Coin('C') # A fair coin toss
>>> density(C).dict
{H: 1/2, T: 1/2}

>>> C2 = Coin('C2', Rational(3, 5)) # An unfair coin
>>> density(C2).dict
{H: 3/5, T: 2/5}


References

R702

https://en.wikipedia.org/wiki/Coin_flipping

sympy.stats.Binomial(name, n, p, succ=1, fail=0)[source]

Create a Finite Random Variable representing a binomial distribution.

Returns a RandomSymbol.

Examples

>>> from sympy.stats import Binomial, density
>>> from sympy import S, Symbol

>>> X = Binomial('X', 4, S.Half) # Four "coin flips"
>>> density(X).dict
{0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16}

>>> n = Symbol('n', positive=True, integer=True)
>>> p = Symbol('p', positive=True)
>>> X = Binomial('X', n, S.Half) # n "coin flips"
>>> density(X).dict
Density(BinomialDistribution(n, 1/2, 1, 0))
>>> density(X).dict.subs(n, 4).doit()
{0: 1/16, 1: 1/4, 2: 3/8, 3: 1/4, 4: 1/16}


References

R703

https://en.wikipedia.org/wiki/Binomial_distribution

R704

http://mathworld.wolfram.com/BinomialDistribution.html

sympy.stats.Hypergeometric(name, N, m, n)[source]

Create a Finite Random Variable representing a hypergeometric distribution.

Returns a RandomSymbol.

Examples

>>> from sympy.stats import Hypergeometric, density
>>> from sympy import S

>>> X = Hypergeometric('X', 10, 5, 3) # 10 marbles, 5 white (success), 3 draws
>>> density(X).dict
{0: 1/12, 1: 5/12, 2: 5/12, 3: 1/12}


References

R705

https://en.wikipedia.org/wiki/Hypergeometric_distribution

R706

http://mathworld.wolfram.com/HypergeometricDistribution.html

sympy.stats.FiniteRV(name, density)[source]

Create a Finite Random Variable given a dict representing the density.

Returns a RandomSymbol.

>>> from sympy.stats import FiniteRV, P, E

>>> density = {0: .1, 1: .2, 2: .3, 3: .4}
>>> X = FiniteRV('X', density)

>>> E(X)
2.00000000000000
>>> P(X >= 2)
0.700000000000000

sympy.stats.Rademacher(name)[source]

Create a Finite Random Variable representing a Rademacher distribution.

Return a RandomSymbol.

Examples

>>> from sympy.stats import Rademacher, density

>>> X = Rademacher('X')
>>> density(X).dict
{-1: 1/2, 1: 1/2}


References

R707

### Discrete Types¶

sympy.stats.Geometric(name, p)[source]

Create a discrete random variable with a Geometric distribution.

The density of the Geometric distribution is given by

$f(k) := p (1 - p)^{k - 1}$
Parameters

p: A probability between 0 and 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Geometric, density, E, variance
>>> from sympy import Symbol, S

>>> p = S.One / 5
>>> z = Symbol("z")

>>> X = Geometric("x", p)

>>> density(X)(z)
(4/5)**(z - 1)/5

>>> E(X)
5

>>> variance(X)
20


References

R708

https://en.wikipedia.org/wiki/Geometric_distribution

R709

http://mathworld.wolfram.com/GeometricDistribution.html

sympy.stats.Poisson(name, lamda)[source]

Create a discrete random variable with a Poisson distribution.

The density of the Poisson distribution is given by

$f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!}$
Parameters

lamda: Positive number, a rate

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Poisson, density, E, variance
>>> from sympy import Symbol, simplify

>>> rate = Symbol("lambda", positive=True)
>>> z = Symbol("z")

>>> X = Poisson("x", rate)

>>> density(X)(z)
lambda**z*exp(-lambda)/factorial(z)

>>> E(X)
lambda

>>> simplify(variance(X))
lambda


References

R710

https://en.wikipedia.org/wiki/Poisson_distribution

R711

http://mathworld.wolfram.com/PoissonDistribution.html

sympy.stats.Logarithmic(name, p)[source]

Create a discrete random variable with a Logarithmic distribution.

The density of the Logarithmic distribution is given by

$f(k) := \frac{-p^k}{k \ln{(1 - p)}}$
Parameters

p: A value between 0 and 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Logarithmic, density, E, variance
>>> from sympy import Symbol, S

>>> p = S.One / 5
>>> z = Symbol("z")

>>> X = Logarithmic("x", p)

>>> density(X)(z)
-5**(-z)/(z*log(4/5))

>>> E(X)
-1/(-4*log(5) + 8*log(2))

>>> variance(X)
-1/((-4*log(5) + 8*log(2))*(-2*log(5) + 4*log(2))) + 1/(-64*log(2)*log(5) + 64*log(2)**2 + 16*log(5)**2) - 10/(-32*log(5) + 64*log(2))


References

R712

https://en.wikipedia.org/wiki/Logarithmic_distribution

R713

http://mathworld.wolfram.com/LogarithmicDistribution.html

sympy.stats.NegativeBinomial(name, r, p)[source]

Create a discrete random variable with a Negative Binomial distribution.

The density of the Negative Binomial distribution is given by

$f(k) := \binom{k + r - 1}{k} (1 - p)^r p^k$
Parameters

r: A positive value

p: A value between 0 and 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import NegativeBinomial, density, E, variance
>>> from sympy import Symbol, S

>>> r = 5
>>> p = S.One / 5
>>> z = Symbol("z")

>>> X = NegativeBinomial("x", r, p)

>>> density(X)(z)
1024*5**(-z)*binomial(z + 4, z)/3125

>>> E(X)
5/4

>>> variance(X)
25/16


References

R714

https://en.wikipedia.org/wiki/Negative_binomial_distribution

R715

http://mathworld.wolfram.com/NegativeBinomialDistribution.html

sympy.stats.YuleSimon(name, rho)[source]

Create a discrete random variable with a Yule-Simon distribution.

The density of the Yule-Simon distribution is given by

$f(k) := \rho B(k, \rho + 1)$
Parameters

rho: A positive value

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import YuleSimon, density, E, variance
>>> from sympy import Symbol, simplify

>>> p = 5
>>> z = Symbol("z")

>>> X = YuleSimon("x", p)

>>> density(X)(z)
5*beta(z, 6)

>>> simplify(E(X))
5/4

>>> simplify(variance(X))
25/48


References

R716

https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution

sympy.stats.Zeta(name, s)[source]

Create a discrete random variable with a Zeta distribution.

The density of the Zeta distribution is given by

$f(k) := \frac{1}{k^s \zeta{(s)}}$
Parameters

s: A value greater than 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Zeta, density, E, variance
>>> from sympy import Symbol

>>> s = 5
>>> z = Symbol("z")

>>> X = Zeta("x", s)

>>> density(X)(z)
1/(z**5*zeta(5))

>>> E(X)
pi**4/(90*zeta(5))

>>> variance(X)
-pi**8/(8100*zeta(5)**2) + zeta(3)/zeta(5)


References

R717

https://en.wikipedia.org/wiki/Zeta_distribution

### Continuous Types¶

sympy.stats.Arcsin(name, a=0, b=1)[source]

Create a Continuous Random Variable with an arcsin distribution.

The density of the arcsin distribution is given by

$f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}}$

with $$x \in (a,b)$$. It must hold that $$-\infty < a < b < \infty$$.

Parameters

a : Real number, the left interval boundary

b : Real number, the right interval boundary

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Arcsin, density, cdf
>>> from sympy import Symbol, simplify

>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")

>>> X = Arcsin("x", a, b)

>>> density(X)(z)
1/(pi*sqrt((-a + z)*(b - z)))

>>> cdf(X)(z)
Piecewise((0, a > z),
(2*asin(sqrt((-a + z)/(-a + b)))/pi, b >= z),
(1, True))


References

R718

https://en.wikipedia.org/wiki/Arcsine_distribution

sympy.stats.Benini(name, alpha, beta, sigma)[source]

Create a Continuous Random Variable with a Benini distribution.

The density of the Benini distribution is given by

$f(x) := e^{-\alpha\log{\frac{x}{\sigma}} -\beta\log^2\left[{\frac{x}{\sigma}}\right]} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)$

This is a heavy-tailed distribution and is also known as the log-Rayleigh distribution.

Parameters

alpha : Real number, $$\alpha > 0$$, a shape

beta : Real number, $$\beta > 0$$, a shape

sigma : Real number, $$\sigma > 0$$, a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Benini, density, cdf
>>> from sympy import Symbol, simplify, pprint

>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")

>>> X = Benini("x", alpha, beta, sigma)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/                  /  z  \\             /  z  \            2/  z  \
|        2*beta*log|-----||  - alpha*log|-----| - beta*log  |-----|
|alpha             \sigma/|             \sigma/             \sigma/
|----- + -----------------|*e
\  z             z        /

>>> cdf(X)(z)
Piecewise((1 - exp(-alpha*log(z/sigma) - beta*log(z/sigma)**2), sigma <= z),
(0, True))


References

R719

https://en.wikipedia.org/wiki/Benini_distribution

R720

http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html

sympy.stats.Beta(name, alpha, beta)[source]

Create a Continuous Random Variable with a Beta distribution.

The density of the Beta distribution is given by

$f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}$

with $$x \in [0,1]$$.

Parameters

alpha : Real number, $$\alpha > 0$$, a shape

beta : Real number, $$\beta > 0$$, a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Beta, density, E, variance
>>> from sympy import Symbol, simplify, pprint, factor

>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")

>>> X = Beta("x", alpha, beta)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1        beta - 1
z         *(1 - z)
--------------------------
B(alpha, beta)

>>> simplify(E(X))
alpha/(alpha + beta)

>>> factor(simplify(variance(X)))
alpha*beta/((alpha + beta)**2*(alpha + beta + 1))


References

R721

https://en.wikipedia.org/wiki/Beta_distribution

R722

sympy.stats.BetaPrime(name, alpha, beta)[source]

Create a continuous random variable with a Beta prime distribution.

The density of the Beta prime distribution is given by

$f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}$

with $$x > 0$$.

Parameters

alpha : Real number, $$\alpha > 0$$, a shape

beta : Real number, $$\beta > 0$$, a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import BetaPrime, density
>>> from sympy import Symbol, pprint

>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")

>>> X = BetaPrime("x", alpha, beta)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1        -alpha - beta
z         *(z + 1)
-------------------------------
B(alpha, beta)


References

R723

https://en.wikipedia.org/wiki/Beta_prime_distribution

R724

http://mathworld.wolfram.com/BetaPrimeDistribution.html

sympy.stats.Cauchy(name, x0, gamma)[source]

Create a continuous random variable with a Cauchy distribution.

The density of the Cauchy distribution is given by

$f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]}$
Parameters

x0 : Real number, the location

gamma : Real number, $$\gamma > 0$$, a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Cauchy, density
>>> from sympy import Symbol

>>> x0 = Symbol("x0")
>>> gamma = Symbol("gamma", positive=True)
>>> z = Symbol("z")

>>> X = Cauchy("x", x0, gamma)

>>> density(X)(z)
1/(pi*gamma*(1 + (-x0 + z)**2/gamma**2))


References

R725

https://en.wikipedia.org/wiki/Cauchy_distribution

R726

http://mathworld.wolfram.com/CauchyDistribution.html

sympy.stats.Chi(name, k)[source]

Create a continuous random variable with a Chi distribution.

The density of the Chi distribution is given by

$f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}$

with $$x \geq 0$$.

Parameters

k : Positive integer, The number of degrees of freedom

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Chi, density, E
>>> from sympy import Symbol, simplify

>>> k = Symbol("k", integer=True)
>>> z = Symbol("z")

>>> X = Chi("x", k)

>>> density(X)(z)
2**(1 - k/2)*z**(k - 1)*exp(-z**2/2)/gamma(k/2)

>>> simplify(E(X))
sqrt(2)*gamma(k/2 + 1/2)/gamma(k/2)


References

R727

https://en.wikipedia.org/wiki/Chi_distribution

R728

http://mathworld.wolfram.com/ChiDistribution.html

sympy.stats.ChiNoncentral(name, k, l)[source]

Create a continuous random variable with a non-central Chi distribution.

The density of the non-central Chi distribution is given by

$f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda} {(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)$

with $$x \geq 0$$. Here, $$I_\nu (x)$$ is the modified Bessel function of the first kind.

Parameters

k : A positive Integer, $$k > 0$$, the number of degrees of freedom

lambda : Real number, $$\lambda > 0$$, Shift parameter

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import ChiNoncentral, density
>>> from sympy import Symbol

>>> k = Symbol("k", integer=True)
>>> l = Symbol("l")
>>> z = Symbol("z")

>>> X = ChiNoncentral("x", k, l)

>>> density(X)(z)
l*z**k*(l*z)**(-k/2)*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)


References

R729

https://en.wikipedia.org/wiki/Noncentral_chi_distribution

sympy.stats.ChiSquared(name, k)[source]

Create a continuous random variable with a Chi-squared distribution.

The density of the Chi-squared distribution is given by

$f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)} x^{\frac{k}{2}-1} e^{-\frac{x}{2}}$

with $$x \geq 0$$.

Parameters

k : Positive integer, The number of degrees of freedom

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import ChiSquared, density, E, variance, moment
>>> from sympy import Symbol

>>> k = Symbol("k", integer=True, positive=True)
>>> z = Symbol("z")

>>> X = ChiSquared("x", k)

>>> density(X)(z)
2**(-k/2)*z**(k/2 - 1)*exp(-z/2)/gamma(k/2)

>>> E(X)
k

>>> variance(X)
2*k

>>> moment(X, 3)
k**3 + 6*k**2 + 8*k


References

R730

https://en.wikipedia.org/wiki/Chi_squared_distribution

R731

http://mathworld.wolfram.com/Chi-SquaredDistribution.html

sympy.stats.Dagum(name, p, a, b)[source]

Create a continuous random variable with a Dagum distribution.

The density of the Dagum distribution is given by

$f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}} {\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right)$

with $$x > 0$$.

Parameters

p : Real number, $$p > 0$$, a shape

a : Real number, $$a > 0$$, a shape

b : Real number, $$b > 0$$, a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Dagum, density, cdf
>>> from sympy import Symbol

>>> p = Symbol("p", positive=True)
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")

>>> X = Dagum("x", p, a, b)

>>> density(X)(z)
a*p*(z/b)**(a*p)*((z/b)**a + 1)**(-p - 1)/z

>>> cdf(X)(z)
Piecewise(((1 + (z/b)**(-a))**(-p), z >= 0), (0, True))


References

R732

https://en.wikipedia.org/wiki/Dagum_distribution

sympy.stats.Erlang(name, k, l)[source]

Create a continuous random variable with an Erlang distribution.

The density of the Erlang distribution is given by

$f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}$

with $$x \in [0,\infty]$$.

Parameters

k : Positive integer

l : Real number, $$\lambda > 0$$, the rate

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Erlang, density, cdf, E, variance
>>> from sympy import Symbol, simplify, pprint

>>> k = Symbol("k", integer=True, positive=True)
>>> l = Symbol("l", positive=True)
>>> z = Symbol("z")

>>> X = Erlang("x", k, l)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k  k - 1  -l*z
l *z     *e
---------------
Gamma(k)

>>> C = cdf(X)(z)
>>> pprint(C, use_unicode=False)
/lowergamma(k, l*z)
|------------------  for z > 0
<     Gamma(k)
|
\        0           otherwise

>>> E(X)
k/l

>>> simplify(variance(X))
k/l**2


References

R733

https://en.wikipedia.org/wiki/Erlang_distribution

R734

http://mathworld.wolfram.com/ErlangDistribution.html

sympy.stats.Exponential(name, rate)[source]

Create a continuous random variable with an Exponential distribution.

The density of the exponential distribution is given by

$f(x) := \lambda \exp(-\lambda x)$

with $$x > 0$$. Note that the expected value is $$1/\lambda$$.

Parameters

rate : A positive Real number, $$\lambda > 0$$, the rate (or inverse scale/inverse mean)

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Exponential, density, cdf, E
>>> from sympy.stats import variance, std, skewness, quantile
>>> from sympy import Symbol

>>> l = Symbol("lambda", positive=True)
>>> z = Symbol("z")
>>> p = Symbol("p")
>>> X = Exponential("x", l)

>>> density(X)(z)
lambda*exp(-lambda*z)

>>> cdf(X)(z)
Piecewise((1 - exp(-lambda*z), z >= 0), (0, True))

>>> quantile(X)(p)
-log(1 - p)/lambda

>>> E(X)
1/lambda

>>> variance(X)
lambda**(-2)

>>> skewness(X)
2

>>> X = Exponential('x', 10)

>>> density(X)(z)
10*exp(-10*z)

>>> E(X)
1/10

>>> std(X)
1/10


References

R735

https://en.wikipedia.org/wiki/Exponential_distribution

R736

http://mathworld.wolfram.com/ExponentialDistribution.html

sympy.stats.FDistribution(name, d1, d2)[source]

Create a continuous random variable with a F distribution.

The density of the F distribution is given by

$f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}} {(d_1 x + d_2)^{d_1 + d_2}}}} {x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)}$

with $$x > 0$$.

Parameters

d1 : $$d_1 > 0$$, where d_1 is the degrees of freedom (n_1 - 1)

d2 : $$d_2 > 0$$, where d_2 is the degrees of freedom (n_2 - 1)

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import FDistribution, density
>>> from sympy import Symbol, simplify, pprint

>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")

>>> X = FDistribution("x", d1, d2)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d2
--    ______________________________
2    /       d1            -d1 - d2
d2  *\/  (d1*z)  *(d1*z + d2)
--------------------------------------
/d1  d2\
z*B|--, --|
\2   2 /


References

R737

https://en.wikipedia.org/wiki/F-distribution

R738

http://mathworld.wolfram.com/F-Distribution.html

sympy.stats.FisherZ(name, d1, d2)[source]

Create a Continuous Random Variable with an Fisher’s Z distribution.

The density of the Fisher’s Z distribution is given by

$f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)} \frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}$
Parameters

d1 : $$d_1 > 0$$, degree of freedom

d2 : $$d_2 > 0$$, degree of freedom

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import FisherZ, density
>>> from sympy import Symbol, simplify, pprint

>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")

>>> X = FisherZ("x", d1, d2)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d1   d2
d1   d2               - -- - --
--   --                 2    2
2    2  /    2*z     \           d1*z
2*d1  *d2  *\d1*e    + d2/         *e
-----------------------------------------
/d1  d2\
B|--, --|
\2   2 /


References

R739

https://en.wikipedia.org/wiki/Fisher%27s_z-distribution

R740

http://mathworld.wolfram.com/Fishersz-Distribution.html

sympy.stats.Frechet(name, a, s=1, m=0)[source]

Create a continuous random variable with a Frechet distribution.

The density of the Frechet distribution is given by

$f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha} e^{-(\frac{x-m}{s})^{-\alpha}}$

with $$x \geq m$$.

Parameters

a : Real number, $$a \in \left(0, \infty\right)$$ the shape

s : Real number, $$s \in \left(0, \infty\right)$$ the scale

m : Real number, $$m \in \left(-\infty, \infty\right)$$ the minimum

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Frechet, density, E, std, cdf
>>> from sympy import Symbol, simplify

>>> a = Symbol("a", positive=True)
>>> s = Symbol("s", positive=True)
>>> m = Symbol("m", real=True)
>>> z = Symbol("z")

>>> X = Frechet("x", a, s, m)

>>> density(X)(z)
a*((-m + z)/s)**(-a - 1)*exp(-((-m + z)/s)**(-a))/s

>>> cdf(X)(z)
Piecewise((exp(-((-m + z)/s)**(-a)), m <= z), (0, True))


References

R741

https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution

sympy.stats.Gamma(name, k, theta)[source]

Create a continuous random variable with a Gamma distribution.

The density of the Gamma distribution is given by

$f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}$

with $$x \in [0,1]$$.

Parameters

k : Real number, $$k > 0$$, a shape

theta : Real number, $$\theta > 0$$, a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Gamma, density, cdf, E, variance
>>> from sympy import Symbol, pprint, simplify

>>> k = Symbol("k", positive=True)
>>> theta = Symbol("theta", positive=True)
>>> z = Symbol("z")

>>> X = Gamma("x", k, theta)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-z
-----
-k  k - 1  theta
theta  *z     *e
---------------------
Gamma(k)

>>> C = cdf(X, meijerg=True)(z)
>>> pprint(C, use_unicode=False)
/            /     z  \
|k*lowergamma|k, -----|
|            \   theta/
<----------------------  for z >= 0
|     Gamma(k + 1)
|
\          0             otherwise

>>> E(X)
k*theta

>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
k*theta


References

R742

https://en.wikipedia.org/wiki/Gamma_distribution

R743

sympy.stats.GammaInverse(name, a, b)[source]

Create a continuous random variable with an inverse Gamma distribution.

The density of the inverse Gamma distribution is given by

$f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp\left(\frac{-\beta}{x}\right)$

with $$x > 0$$.

Parameters

a : Real number, $$a > 0$$ a shape

b : Real number, $$b > 0$$ a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import GammaInverse, density, cdf, E, variance
>>> from sympy import Symbol, pprint

>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")

>>> X = GammaInverse("x", a, b)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-b
---
a  -a - 1   z
b *z      *e
---------------
Gamma(a)

>>> cdf(X)(z)
Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True))


References

R744

https://en.wikipedia.org/wiki/Inverse-gamma_distribution

sympy.stats.Gompertz(name, b, eta)[source]

Create a Continuous Random Variable with Gompertz distribution.

The density of the Gompertz distribution is given by

$f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right)$

with :math: ‘x in [0, inf)’.

Parameters

b: Real number, ‘b > 0’ a scale

eta: Real number, ‘eta > 0’ a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Gompertz, density, E, variance
>>> from sympy import Symbol, simplify, pprint

>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> z = Symbol("z")

>>> X = Gompertz("x", b, eta)

>>> density(X)(z)
b*eta*exp(eta)*exp(b*z)*exp(-eta*exp(b*z))


References

R745

https://en.wikipedia.org/wiki/Gompertz_distribution

sympy.stats.Gumbel(name, beta, mu, minimum=False)[source]

Create a Continuous Random Variable with Gumbel distribution.

The density of the Gumbel distribution is given by

For Maximum

$f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta} - \exp \left( -\dfrac{x - \mu}{\beta} \right) \right)$

with $$x \in [ - \infty, \infty ]$$.

For Minimum

$f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta}$

with $$x \in [ - \infty, \infty ]$$.

Parameters

mu : Real number, ‘mu’ is a location

beta : Real number, ‘beta > 0’ is a scale

minimum : Boolean, by default, False, set to True for enabling minimum distribution

Returns

A RandomSymbol

Examples

>>> from sympy.stats import Gumbel, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint
>>> x = Symbol("x")
>>> mu = Symbol("mu")
>>> beta = Symbol("beta", positive=True)
>>> X = Gumbel("x", beta, mu)
>>> density(X)(x)
exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta
>>> cdf(X)(x)
exp(-exp(-(-mu + x)/beta))


References

R746

http://mathworld.wolfram.com/GumbelDistribution.html

R747

https://en.wikipedia.org/wiki/Gumbel_distribution

R748

http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html

R749

http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html

sympy.stats.Kumaraswamy(name, a, b)[source]

Create a Continuous Random Variable with a Kumaraswamy distribution.

The density of the Kumaraswamy distribution is given by

$f(x) := a b x^{a-1} (1-x^a)^{b-1}$

with $$x \in [0,1]$$.

Parameters

a : Real number, $$a > 0$$ a shape

b : Real number, $$b > 0$$ a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Kumaraswamy, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint

>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")

>>> X = Kumaraswamy("x", a, b)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
b - 1
a - 1 /     a\
a*b*z     *\1 - z /

>>> cdf(X)(z)
Piecewise((0, z < 0), (1 - (1 - z**a)**b, z <= 1), (1, True))


References

R750

https://en.wikipedia.org/wiki/Kumaraswamy_distribution

sympy.stats.Laplace(name, mu, b)[source]

Create a continuous random variable with a Laplace distribution.

The density of the Laplace distribution is given by

$f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right)$
Parameters

mu : Real number or a list/matrix, the location (mean) or the

location vector

b : Real number or a positive definite matrix, representing a scale

or the covariance matrix.

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Laplace, density, cdf
>>> from sympy import Symbol, pprint

>>> mu = Symbol("mu")
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")

>>> X = Laplace("x", mu, b)

>>> density(X)(z)
exp(-Abs(mu - z)/b)/(2*b)

>>> cdf(X)(z)
Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True))

>>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]])
>>> pprint(density(L)(1, 2), use_unicode=False)
5        /     ____\
e *besselk\0, \/ 35 /
---------------------
pi


References

R751

https://en.wikipedia.org/wiki/Laplace_distribution

R752

http://mathworld.wolfram.com/LaplaceDistribution.html

sympy.stats.Logistic(name, mu, s)[source]

Create a continuous random variable with a logistic distribution.

The density of the logistic distribution is given by

$f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}$
Parameters

mu : Real number, the location (mean)

s : Real number, $$s > 0$$ a scale

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Logistic, density, cdf
>>> from sympy import Symbol

>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")

>>> X = Logistic("x", mu, s)

>>> density(X)(z)
exp((mu - z)/s)/(s*(exp((mu - z)/s) + 1)**2)

>>> cdf(X)(z)
1/(exp((mu - z)/s) + 1)


References

R753

https://en.wikipedia.org/wiki/Logistic_distribution

R754

http://mathworld.wolfram.com/LogisticDistribution.html

sympy.stats.LogNormal(name, mean, std)[source]

Create a continuous random variable with a log-normal distribution.

The density of the log-normal distribution is given by

$f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}} e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}$

with $$x \geq 0$$.

Parameters

mu : Real number, the log-scale

sigma : Real number, $$\sigma^2 > 0$$ a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import LogNormal, density
>>> from sympy import Symbol, simplify, pprint

>>> mu = Symbol("mu", real=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")

>>> X = LogNormal("x", mu, sigma)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-(-mu + log(z))
-----------------
2
___      2*sigma
\/ 2 *e
------------------------
____
2*\/ pi *sigma*z

>>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1

>>> density(X)(z)
sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z)


References

R755

https://en.wikipedia.org/wiki/Lognormal

R756

http://mathworld.wolfram.com/LogNormalDistribution.html

sympy.stats.Maxwell(name, a)[source]

Create a continuous random variable with a Maxwell distribution.

The density of the Maxwell distribution is given by

$f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}$

with $$x \geq 0$$.

Parameters

a : Real number, $$a > 0$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Maxwell, density, E, variance
>>> from sympy import Symbol, simplify

>>> a = Symbol("a", positive=True)
>>> z = Symbol("z")

>>> X = Maxwell("x", a)

>>> density(X)(z)
sqrt(2)*z**2*exp(-z**2/(2*a**2))/(sqrt(pi)*a**3)

>>> E(X)
2*sqrt(2)*a/sqrt(pi)

>>> simplify(variance(X))
a**2*(-8 + 3*pi)/pi


References

R757

https://en.wikipedia.org/wiki/Maxwell_distribution

R758

http://mathworld.wolfram.com/MaxwellDistribution.html

sympy.stats.Nakagami(name, mu, omega)[source]

Create a continuous random variable with a Nakagami distribution.

The density of the Nakagami distribution is given by

$f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1} \exp\left(-\frac{\mu}{\omega}x^2 \right)$

with $$x > 0$$.

Parameters

mu : Real number, $$\mu \geq \frac{1}{2}$$ a shape

omega : Real number, $$\omega > 0$$, the spread

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Nakagami, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint

>>> mu = Symbol("mu", positive=True)
>>> omega = Symbol("omega", positive=True)
>>> z = Symbol("z")

>>> X = Nakagami("x", mu, omega)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-mu*z
-------
mu      -mu  2*mu - 1  omega
2*mu  *omega   *z        *e
----------------------------------
Gamma(mu)

>>> simplify(E(X))
sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1)

>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
omega*Gamma (mu + 1/2)
omega - -----------------------
Gamma(mu)*Gamma(mu + 1)

>>> cdf(X)(z)
Piecewise((lowergamma(mu, mu*z**2/omega)/gamma(mu), z > 0),
(0, True))


References

R759

https://en.wikipedia.org/wiki/Nakagami_distribution

sympy.stats.Normal(name, mean, std)[source]

Create a continuous random variable with a Normal distribution.

The density of the Normal distribution is given by

$f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }$
Parameters

mu : Real number or a list representing the mean or the mean vector

sigma : Real number or a positive definite square matrix,

$$\sigma^2 > 0$$ the variance

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile
>>> from sympy import Symbol, simplify, pprint, factor, together, factor_terms

>>> mu = Symbol("mu")
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> y = Symbol("y")
>>> p = Symbol("p")
>>> X = Normal("x", mu, sigma)

>>> density(X)(z)
sqrt(2)*exp(-(-mu + z)**2/(2*sigma**2))/(2*sqrt(pi)*sigma)

>>> C = simplify(cdf(X))(z) # it needs a little more help...
>>> pprint(C, use_unicode=False)
/  ___          \
|\/ 2 *(-mu + z)|
erf|---------------|
\    2*sigma    /   1
-------------------- + -
2             2

>>> quantile(X)(p)
mu + sqrt(2)*sigma*erfinv(2*p - 1)

>>> simplify(skewness(X))
0

>>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1
>>> density(X)(z)
sqrt(2)*exp(-z**2/2)/(2*sqrt(pi))

>>> E(2*X + 1)
1

>>> simplify(std(2*X + 1))
2

>>> m = Normal('X', [1, 2], [[2, 1], [1, 2]])
>>> from sympy.stats.joint_rv import marginal_distribution
>>> pprint(density(m)(y, z), use_unicode=False)
/1   y\ /2*y   z\   /    z\ /  y   2*z    \
|- - -|*|--- - -| + |1 - -|*|- - + --- - 1|
___  \2   2/ \ 3    3/   \    2/ \  3    3     /
\/ 3 *e
--------------------------------------------------
6*pi

>>> marginal_distribution(m, m[0])(1)
1/(2*sqrt(pi))


References

R760

https://en.wikipedia.org/wiki/Normal_distribution

R761

http://mathworld.wolfram.com/NormalDistributionFunction.html

sympy.stats.Pareto(name, xm, alpha)[source]

Create a continuous random variable with the Pareto distribution.

The density of the Pareto distribution is given by

$f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}}$

with $$x \in [x_m,\infty]$$.

Parameters

xm : Real number, $$x_m > 0$$, a scale

alpha : Real number, $$\alpha > 0$$, a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Pareto, density
>>> from sympy import Symbol

>>> xm = Symbol("xm", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")

>>> X = Pareto("x", xm, beta)

>>> density(X)(z)
beta*xm**beta*z**(-beta - 1)


References

R762

https://en.wikipedia.org/wiki/Pareto_distribution

R763

http://mathworld.wolfram.com/ParetoDistribution.html

sympy.stats.QuadraticU(name, a, b)[source]

Create a Continuous Random Variable with a U-quadratic distribution.

The density of the U-quadratic distribution is given by

$f(x) := \alpha (x-\beta)^2$

with $$x \in [a,b]$$.

Parameters

a : Real number

b : Real number, $$a < b$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import QuadraticU, density, E, variance
>>> from sympy import Symbol, simplify, factor, pprint

>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")

>>> X = QuadraticU("x", a, b)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/                2
|   /  a   b    \
|12*|- - - - + z|
|   \  2   2    /
<-----------------  for And(b >= z, a <= z)
|            3
|    (-a + b)
|
\        0                 otherwise


References

R764

sympy.stats.RaisedCosine(name, mu, s)[source]

Create a Continuous Random Variable with a raised cosine distribution.

The density of the raised cosine distribution is given by

$f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right)$

with $$x \in [\mu-s,\mu+s]$$.

Parameters

mu : Real number

s : Real number, $$s > 0$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import RaisedCosine, density, E, variance
>>> from sympy import Symbol, simplify, pprint

>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")

>>> X = RaisedCosine("x", mu, s)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/   /pi*(-mu + z)\
|cos|------------| + 1
|   \     s      /
<---------------------  for And(z >= mu - s, z <= mu + s)
|         2*s
|
\          0                        otherwise


References

R765

https://en.wikipedia.org/wiki/Raised_cosine_distribution

sympy.stats.Rayleigh(name, sigma)[source]

Create a continuous random variable with a Rayleigh distribution.

The density of the Rayleigh distribution is given by

$f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}$

with $$x > 0$$.

Parameters

sigma : Real number, $$\sigma > 0$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Rayleigh, density, E, variance
>>> from sympy import Symbol, simplify

>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")

>>> X = Rayleigh("x", sigma)

>>> density(X)(z)
z*exp(-z**2/(2*sigma**2))/sigma**2

>>> E(X)
sqrt(2)*sqrt(pi)*sigma/2

>>> variance(X)
-pi*sigma**2/2 + 2*sigma**2


References

R766

https://en.wikipedia.org/wiki/Rayleigh_distribution

R767

http://mathworld.wolfram.com/RayleighDistribution.html

sympy.stats.StudentT(name, nu)[source]

Create a continuous random variable with a student’s t distribution.

The density of the student’s t distribution is given by

$f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}$
Parameters

nu : Real number, $$\nu > 0$$, the degrees of freedom

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import StudentT, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint

>>> nu = Symbol("nu", positive=True)
>>> z = Symbol("z")

>>> X = StudentT("x", nu)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
nu   1
- -- - -
2    2
/     2\
|    z |
|1 + --|
\    nu/
-----------------
____  /     nu\
\/ nu *B|1/2, --|
\     2 /

>>> cdf(X)(z)
1/2 + z*gamma(nu/2 + 1/2)*hyper((1/2, nu/2 + 1/2), (3/2,),
-z**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2))


References

R768

https://en.wikipedia.org/wiki/Student_t-distribution

R769

http://mathworld.wolfram.com/Studentst-Distribution.html

sympy.stats.ShiftedGompertz(name, b, eta)[source]

Create a continuous random variable with a Shifted Gompertz distribution.

The density of the Shifted Gompertz distribution is given by

$f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right]$

with :math: ‘x in [0, inf)’.

Parameters

b: Real number, ‘b > 0’ a scale

eta: Real number, ‘eta > 0’ a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import ShiftedGompertz, density, E, variance
>>> from sympy import Symbol

>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> x = Symbol("x")

>>> X = ShiftedGompertz("x", b, eta)

>>> density(X)(x)
b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x))


References

R770

https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution

sympy.stats.Trapezoidal(name, a, b, c, d)[source]

Create a continuous random variable with a trapezoidal distribution.

The density of the trapezoidal distribution is given by

$\begin{split}f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\ \frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\ \frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\ 0 & \mathrm{for\ } d < x. \end{cases}\end{split}$
Parameters

a : Real number, $$a < d$$

b : Real number, $$a <= b < c$$

c : Real number, $$b < c <= d$$

d : Real number

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Trapezoidal, density, E
>>> from sympy import Symbol, pprint

>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> d = Symbol("d")
>>> z = Symbol("z")

>>> X = Trapezoidal("x", a,b,c,d)

>>> pprint(density(X)(z), use_unicode=False)
/        -2*a + 2*z
|-------------------------  for And(a <= z, b > z)
|(-a + b)*(-a - b + c + d)
|
|           2
|     --------------        for And(b <= z, c > z)
<     -a - b + c + d
|
|        2*d - 2*z
|-------------------------  for And(d >= z, c <= z)
|(-c + d)*(-a - b + c + d)
|
\            0                     otherwise


References

R771

https://en.wikipedia.org/wiki/Trapezoidal_distribution

sympy.stats.Triangular(name, a, b, c)[source]

Create a continuous random variable with a triangular distribution.

The density of the triangular distribution is given by

$\begin{split}f(x) := \begin{cases} 0 & \mathrm{for\ } x < a, \\ \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\ \frac{2}{b-a} & \mathrm{for\ } x = c, \\ \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\ 0 & \mathrm{for\ } b < x. \end{cases}\end{split}$
Parameters

a : Real number, $$a \in \left(-\infty, \infty\right)$$

b : Real number, $$a < b$$

c : Real number, $$a \leq c \leq b$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Triangular, density, E
>>> from sympy import Symbol, pprint

>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> z = Symbol("z")

>>> X = Triangular("x", a,b,c)

>>> pprint(density(X)(z), use_unicode=False)
/    -2*a + 2*z
|-----------------  for And(a <= z, c > z)
|(-a + b)*(-a + c)
|
|       2
|     ------              for c = z
<     -a + b
|
|   2*b - 2*z
|----------------   for And(b >= z, c < z)
|(-a + b)*(b - c)
|
\        0                otherwise


References

R772

https://en.wikipedia.org/wiki/Triangular_distribution

R773

http://mathworld.wolfram.com/TriangularDistribution.html

sympy.stats.Uniform(name, left, right)[source]

Create a continuous random variable with a uniform distribution.

The density of the uniform distribution is given by

$\begin{split}f(x) := \begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases}\end{split}$

with $$x \in [a,b]$$.

Parameters

a : Real number, $$-\infty < a$$ the left boundary

b : Real number, $$a < b < \infty$$ the right boundary

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Uniform, density, cdf, E, variance, skewness
>>> from sympy import Symbol, simplify

>>> a = Symbol("a", negative=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")

>>> X = Uniform("x", a, b)

>>> density(X)(z)
Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True))

>>> cdf(X)(z)
Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True))

>>> E(X)
a/2 + b/2

>>> simplify(variance(X))
a**2/12 - a*b/6 + b**2/12


References

R774

https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29

R775

http://mathworld.wolfram.com/UniformDistribution.html

sympy.stats.UniformSum(name, n)[source]

Create a continuous random variable with an Irwin-Hall distribution.

The probability distribution function depends on a single parameter $$n$$ which is an integer.

The density of the Irwin-Hall distribution is given by

$f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\left\lfloor x\right\rfloor}(-1)^k \binom{n}{k}(x-k)^{n-1}$
Parameters

n : A positive Integer, $$n > 0$$

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import UniformSum, density, cdf
>>> from sympy import Symbol, pprint

>>> n = Symbol("n", integer=True)
>>> z = Symbol("z")

>>> X = UniformSum("x", n)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
floor(z)
___
\
\         k         n - 1 /n\
)    (-1) *(-k + z)     *| |
/                         \k/
/__,
k = 0
--------------------------------
(n - 1)!

>>> cdf(X)(z)
Piecewise((0, z < 0), (Sum((-1)**_k*(-_k + z)**n*binomial(n, _k),
(_k, 0, floor(z)))/factorial(n), n >= z), (1, True))


Compute cdf with specific ‘x’ and ‘n’ values as follows : >>> cdf(UniformSum(“x”, 5), evaluate=False)(2).doit() 9/40

The argument evaluate=False prevents an attempt at evaluation of the sum for general n, before the argument 2 is passed.

References

R776

https://en.wikipedia.org/wiki/Uniform_sum_distribution

R777

http://mathworld.wolfram.com/UniformSumDistribution.html

sympy.stats.VonMises(name, mu, k)[source]

Create a Continuous Random Variable with a von Mises distribution.

The density of the von Mises distribution is given by

$f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}$

with $$x \in [0,2\pi]$$.

Parameters

mu : Real number, measure of location

k : Real number, measure of concentration

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import VonMises, density, E, variance
>>> from sympy import Symbol, simplify, pprint

>>> mu = Symbol("mu")
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")

>>> X = VonMises("x", mu, k)

>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k*cos(mu - z)
e
------------------
2*pi*besseli(0, k)


References

R778

https://en.wikipedia.org/wiki/Von_Mises_distribution

R779

http://mathworld.wolfram.com/vonMisesDistribution.html

sympy.stats.Weibull(name, alpha, beta)[source]

Create a continuous random variable with a Weibull distribution.

The density of the Weibull distribution is given by

$\begin{split}f(x) := \begin{cases} \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^{k}} & x\geq0\\ 0 & x<0 \end{cases}\end{split}$
Parameters

lambda : Real number, $$\lambda > 0$$ a scale

k : Real number, $$k > 0$$ a shape

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import Weibull, density, E, variance
>>> from sympy import Symbol, simplify

>>> l = Symbol("lambda", positive=True)
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")

>>> X = Weibull("x", l, k)

>>> density(X)(z)
k*(z/lambda)**(k - 1)*exp(-(z/lambda)**k)/lambda

>>> simplify(E(X))
lambda*gamma(1 + 1/k)

>>> simplify(variance(X))
lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k))


References

R780

https://en.wikipedia.org/wiki/Weibull_distribution

R781

http://mathworld.wolfram.com/WeibullDistribution.html

sympy.stats.WignerSemicircle(name, R)[source]

Create a continuous random variable with a Wigner semicircle distribution.

The density of the Wigner semicircle distribution is given by

$f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2}$

with $$x \in [-R,R]$$.

Parameters

R : Real number, $$R > 0$$, the radius

Returns

A $$RandomSymbol$$.

Examples

>>> from sympy.stats import WignerSemicircle, density, E
>>> from sympy import Symbol, simplify

>>> R = Symbol("R", positive=True)
>>> z = Symbol("z")

>>> X = WignerSemicircle("x", R)

>>> density(X)(z)
2*sqrt(R**2 - z**2)/(pi*R**2)

>>> E(X)
0


References

R782

https://en.wikipedia.org/wiki/Wigner_semicircle_distribution

R783

sympy.stats.ContinuousRV(symbol, density, set=Interval(-oo, oo))[source]

Create a Continuous Random Variable given the following:

– a symbol – a probability density function – set on which the pdf is valid (defaults to entire real line)

Returns a RandomSymbol.

Many common continuous random variable types are already implemented. This function should be necessary only very rarely.

Examples

>>> from sympy import Symbol, sqrt, exp, pi
>>> from sympy.stats import ContinuousRV, P, E

>>> x = Symbol("x")

>>> pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
>>> X = ContinuousRV(x, pdf)

>>> E(X)
0
>>> P(X>0)
1/2


### Joint Types¶

sympy.stats.JointRV(symbol, pdf, _set=None)[source]

Create a Joint Random Variable where each of its component is conitinuous, given the following:

– a symbol – a PDF in terms of indexed symbols of the symbol given as the first argument

NOTE: As of now, the set for each component for a $$JointRV$$ is equal to the set of all integers, which can not be changed.

Returns a RandomSymbol.

Examples

>>> from sympy import symbols, exp, pi, Indexed, S
>>> from sympy.stats import density
>>> from sympy.stats.joint_rv_types import JointRV

>>> x1, x2 = (Indexed('x', i) for i in (1, 2))
>>> pdf = exp(-x1**2/2 + x1 - x2**2/2 - S(1)/2)/(2*pi)

>>> N1 = JointRV('x', pdf) #Multivariate Normal distribution
>>> density(N1)(1, 2)
exp(-2)/(2*pi)

sympy.stats.GeneralizedMultivariateLogGamma(syms, delta, v, lamda, mu)[source]

Creates a joint random variable with generalized multivariate log gamma distribution.

The joint pdf can be found at [1].

Parameters

syms: list/tuple/set of symbols for identifying each component

delta: A constant in range [0, 1]

v: positive real

lamda: a list of positive reals

mu: a list of positive reals

Returns

A Random Symbol

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma
>>> from sympy import symbols, S
>>> v = 1
>>> l, mu = [1, 1, 1], [1, 1, 1]
>>> d = S.Half
>>> y = symbols('y_1:4', positive=True)
>>> Gd = GeneralizedMultivariateLogGamma('G', d, v, l, mu)
>>> density(Gd)(y[0], y[1], y[2])
Sum(2**(-n)*exp((n + 1)*(y_1 + y_2 + y_3) - exp(y_1) - exp(y_2) -
exp(y_3))/gamma(n + 1)**3, (n, 0, oo))/2


Note

If the GeneralizedMultivariateLogGamma is too long to type use, $$from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGamma as GMVLG$$ If you want to pass the matrix omega instead of the constant delta, then use, GeneralizedMultivariateLogGammaOmega.

References

R784

https://en.wikipedia.org/wiki/Generalized_multivariate_log-gamma_distribution

R785

https://www.researchgate.net/publication/234137346_On_a_multivariate_log-gamma_distribution_and_the_use_of_the_distribution_in_the_Bayesian_analysis

sympy.stats.GeneralizedMultivariateLogGammaOmega(syms, omega, v, lamda, mu)[source]

Extends GeneralizedMultivariateLogGamma.

Parameters

syms: list/tuple/set of symbols for identifying each component

omega: A square matrix

Every element of square matrix must be absolute value of square root of correlation coefficient

v: positive real

lamda: a list of positive reals

mu: a list of positive reals

Returns

A Random Symbol

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega
>>> from sympy import Matrix, symbols, S
>>> omega = Matrix([[1, S.Half, S.Half], [S.Half, 1, S.Half], [S.Half, S.Half, 1]])
>>> v = 1
>>> l, mu = [1, 1, 1], [1, 1, 1]
>>> G = GeneralizedMultivariateLogGammaOmega('G', omega, v, l, mu)
>>> y = symbols('y_1:4', positive=True)
>>> density(G)(y[0], y[1], y[2])
sqrt(2)*Sum((1 - sqrt(2)/2)**n*exp((n + 1)*(y_1 + y_2 + y_3) - exp(y_1) -
exp(y_2) - exp(y_3))/gamma(n + 1)**3, (n, 0, oo))/2


Notes

If the GeneralizedMultivariateLogGammaOmega is too long to type use, $$from sympy.stats.joint_rv_types import GeneralizedMultivariateLogGammaOmega as GMVLGO$$

References

See references of GeneralizedMultivariateLogGamma.

sympy.stats.Multinomial(syms, n, *p)[source]

Creates a discrete random variable with Multinomial Distribution.

The density of the said distribution can be found at [1].

Parameters

n: positive integer of class Integer,

number of trials

p: event probabilites, >= 0 and <= 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import Multinomial
>>> from sympy import symbols
>>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True)
>>> p1, p2, p3 = symbols('p1, p2, p3', positive=True)
>>> M = Multinomial('M', 3, p1, p2, p3)
>>> density(M)(x1, x2, x3)
Piecewise((6*p1**x1*p2**x2*p3**x3/(factorial(x1)*factorial(x2)*factorial(x3)),
Eq(x1 + x2 + x3, 3)), (0, True))
>>> marginal_distribution(M, M[0])(x1).subs(x1, 1)
3*p1*p2**2 + 6*p1*p2*p3 + 3*p1*p3**2


References

R786

https://en.wikipedia.org/wiki/Multinomial_distribution

R787

http://mathworld.wolfram.com/MultinomialDistribution.html

sympy.stats.MultivariateBeta(syms, *alpha)[source]

Creates a continuous random variable with Dirichlet/Multivariate Beta Distribution.

The density of the dirichlet distribution can be found at [1].

Parameters

alpha: positive real numbers signifying concentration numbers.

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import MultivariateBeta
>>> from sympy import Symbol
>>> a1 = Symbol('a1', positive=True)
>>> a2 = Symbol('a2', positive=True)
>>> B = MultivariateBeta('B', [a1, a2])
>>> C = MultivariateBeta('C', a1, a2)
>>> x = Symbol('x')
>>> y = Symbol('y')
>>> density(B)(x, y)
x**(a1 - 1)*y**(a2 - 1)*gamma(a1 + a2)/(gamma(a1)*gamma(a2))
>>> marginal_distribution(C, C[0])(x)
x**(a1 - 1)*gamma(a1 + a2)/(a2*gamma(a1)*gamma(a2))


References

R788

https://en.wikipedia.org/wiki/Dirichlet_distribution

R789

http://mathworld.wolfram.com/DirichletDistribution.html

sympy.stats.MultivariateEwens(syms, n, theta)[source]

Creates a discrete random variable with Multivariate Ewens Distribution.

The density of the said distribution can be found at [1].

Parameters

n: positive integer of class Integer,

size of the sample or the integer whose partitions are considered

theta: mutation rate, must be positive real number.

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import MultivariateEwens
>>> from sympy import Symbol
>>> a1 = Symbol('a1', positive=True)
>>> a2 = Symbol('a2', positive=True)
>>> ed = MultivariateEwens('E', 2, 1)
>>> density(ed)(a1, a2)
Piecewise((2**(-a2)/(factorial(a1)*factorial(a2)), Eq(a1 + 2*a2, 2)), (0, True))
>>> marginal_distribution(ed, ed[0])(a1)
Piecewise((1/factorial(a1), Eq(a1, 2)), (0, True))


References

R790

https://en.wikipedia.org/wiki/Ewens%27s_sampling_formula

R791

http://www.stat.rutgers.edu/home/hcrane/Papers/STS529.pdf

sympy.stats.MultivariateT(syms, mu, sigma, v)[source]

Creates a joint random variable with multivariate T-distribution.

Parameters

syms: list/tuple/set of symbols for identifying each component

mu: A list/tuple/set consisting of k means,represents a k

dimensional location vector

sigma: The shape matrix for the distribution

Returns

A random symbol

sympy.stats.NegativeMultinomial(syms, k0, *p)[source]

Creates a discrete random variable with Negative Multinomial Distribution.

The density of the said distribution can be found at [1].

Parameters

k0: positive integer of class Integer,

number of failures before the experiment is stopped

p: event probabilites, >= 0 and <= 1

Returns

A RandomSymbol.

Examples

>>> from sympy.stats import density
>>> from sympy.stats.joint_rv import marginal_distribution
>>> from sympy.stats.joint_rv_types import NegativeMultinomial
>>> from sympy import symbols
>>> x1, x2, x3 = symbols('x1, x2, x3', nonnegative=True, integer=True)
>>> p1, p2, p3 = symbols('p1, p2, p3', positive=True)
>>> N = NegativeMultinomial('M', 3, p1, p2, p3)
>>> N_c = NegativeMultinomial('M', 3, 0.1, 0.1, 0.1)
>>> density(N)(x1, x2, x3)
p1**x1*p2**x2*p3**x3*(-p1 - p2 - p3 + 1)**3*gamma(x1 + x2 +
x3 + 3)/(2*factorial(x1)*factorial(x2)*factorial(x3))
>>> marginal_distribution(N_c, N_c[0])(1).evalf().round(2)
0.25


References

R792

https://en.wikipedia.org/wiki/Negative_multinomial_distribution

R793

http://mathworld.wolfram.com/NegativeBinomialDistribution.html

sympy.stats.NormalGamma(syms, mu, lamda, alpha, beta)[source]

Creates a bivariate joint random variable with multivariate Normal gamma distribution.

Parameters

syms: list/tuple/set of two symbols for identifying each component

mu: A real number, as the mean of the normal distribution

alpha: a positive integer

beta: a positive integer

lamda: a positive integer

Returns

A random symbol

### Stochastic Processes¶

sympy.stats.DiscreteMarkovChain(sym, state_space=Reals, trans_probs=None)[source]

Represents discrete time Markov chain.

Parameters

sym: Symbol/string_types

state_space: Set

Optional, by default, S.Reals

trans_probs: Matrix/ImmutableMatrix/MatrixSymbol

Optional, by default, None

Examples

>>> from sympy.stats import DiscreteMarkovChain, TransitionMatrixOf
>>> from sympy import Matrix, MatrixSymbol, Eq
>>> from sympy.stats import P
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> YS = DiscreteMarkovChain("Y")
>>> Y.state_space
FiniteSet(0, 1, 2)
>>> Y.transition_probabilities
Matrix([
[0.5, 0.2, 0.3],
[0.2, 0.5, 0.3],
[0.2, 0.3, 0.5]])
>>> TS = MatrixSymbol('T', 3, 3)
>>> P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TS))
T[0, 2]*T[1, 0] + T[1, 1]*T[1, 2] + T[1, 2]*T[2, 2]
>>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2)
0.36


References

R794

https://en.wikipedia.org/wiki/Markov_chain#Discrete-time_Markov_chain

R795

https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf

## Interface¶

sympy.stats.P(condition, given_condition=None, numsamples=None, evaluate=True, **kwargs)[source]

Probability that a condition is true, optionally given a second condition

Parameters

condition : Combination of Relationals containing RandomSymbols

The condition of which you want to compute the probability

given_condition : Combination of Relationals containing RandomSymbols

A conditional expression. P(X > 1, X > 0) is expectation of X > 1 given X > 0

numsamples : int

Enables sampling and approximates the probability with this many samples

evaluate : Bool (defaults to True)

In case of continuous systems return unevaluated integral

Examples

>>> from sympy.stats import P, Die
>>> from sympy import Eq
>>> X, Y = Die('X', 6), Die('Y', 6)
>>> P(X > 3)
1/2
>>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2
1/4
>>> P(X > Y)
5/12

class sympy.stats.Probability[source]

Symbolic expression for the probability.

Examples

>>> from sympy.stats import Probability, Normal
>>> from sympy import Integral
>>> X = Normal("X", 0, 1)
>>> prob = Probability(X > 1)
>>> prob
Probability(X > 1)


Integral representation:

>>> prob.rewrite(Integral)
Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo))


Evaluation of the integral:

>>> prob.evaluate_integral()
sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi))

sympy.stats.E(expr, condition=None, numsamples=None, evaluate=True, **kwargs)[source]

Returns the expected value of a random expression

Parameters

expr : Expr containing RandomSymbols

The expression of which you want to compute the expectation value

given : Expr containing RandomSymbols

A conditional expression. E(X, X>0) is expectation of X given X > 0

numsamples : int

Enables sampling and approximates the expectation with this many samples

evalf : Bool (defaults to True)

If sampling return a number rather than a complex expression

evaluate : Bool (defaults to True)

In case of continuous systems return unevaluated integral

Examples

>>> from sympy.stats import E, Die
>>> X = Die('X', 6)
>>> E(X)
7/2
>>> E(2*X + 1)
8

>>> E(X, X > 3) # Expectation of X given that it is above 3
5

class sympy.stats.Expectation[source]

Symbolic expression for the expectation.

Examples

>>> from sympy.stats import Expectation, Normal, Probability
>>> from sympy import symbols, Integral
>>> mu = symbols("mu")
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Expectation(X)
Expectation(X)
>>> Expectation(X).evaluate_integral().simplify()
mu


To get the integral expression of the expectation:

>>> Expectation(X).rewrite(Integral)
Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))


The same integral expression, in more abstract terms:

>>> Expectation(X).rewrite(Probability)
Integral(x*Probability(Eq(X, x)), (x, -oo, oo))


This class is aware of some properties of the expectation:

>>> from sympy.abc import a
>>> Expectation(a*X)
Expectation(a*X)
>>> Y = Normal("Y", 0, 1)
>>> Expectation(X + Y)
Expectation(X + Y)


To expand the Expectation into its expression, use doit():

>>> Expectation(X + Y).doit()
Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y).doit()
a*Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y)
Expectation(a*X + Y)

sympy.stats.density(expr, condition=None, evaluate=True, numsamples=None, **kwargs)[source]

Probability density of a random expression, optionally given a second condition.

This density will take on different forms for different types of probability spaces. Discrete variables produce Dicts. Continuous variables produce Lambdas.

Parameters

expr : Expr containing RandomSymbols

The expression of which you want to compute the density value

condition : Relational containing RandomSymbols

A conditional expression. density(X > 1, X > 0) is density of X > 1 given X > 0

numsamples : int

Enables sampling and approximates the density with this many samples

Examples

>>> from sympy.stats import density, Die, Normal
>>> from sympy import Symbol

>>> x = Symbol('x')
>>> D = Die('D', 6)
>>> X = Normal(x, 0, 1)

>>> density(D).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
>>> density(2*D).dict
{2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}
>>> density(X)(x)
sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))

sympy.stats.entropy(expr, condition=None, **kwargs)[source]

Calculuates entropy of a probability distribution

Parameters

expression : the random expression whose entropy is to be calculated

condition : optional, to specify conditions on random expression

b: base of the logarithm, optional

By default, it is taken as Euler’s number

Returns

result : Entropy of the expression, a constant

Examples

>>> from sympy.stats import Normal, Die, entropy
>>> X = Normal('X', 0, 1)
>>> entropy(X)
log(2)/2 + 1/2 + log(pi)/2

>>> D = Die('D', 4)
>>> entropy(D)
log(4)


References

R796
R797

https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf

R798

sympy.stats.given(expr, condition=None, **kwargs)[source]

Conditional Random Expression From a random expression and a condition on that expression creates a new probability space from the condition and returns the same expression on that conditional probability space.

Examples

>>> from sympy.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}


Following convention, if the condition is a random symbol then that symbol is considered fixed.

>>> from sympy.stats import Normal
>>> from sympy import pprint
>>> from sympy.abc import z

>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___       2
\/ 2 *e
------------------
____
2*\/ pi

sympy.stats.where(condition, given_condition=None, **kwargs)[source]

Returns the domain where a condition is True.

Examples

>>> from sympy.stats import where, Die, Normal
>>> from sympy import symbols, And

>>> D1, D2 = Die('a', 6), Die('b', 6)
>>> a, b = D1.symbol, D2.symbol
>>> X = Normal('x', 0, 1)

>>> where(X**2<1)
Domain: (-1 < x) & (x < 1)

>>> where(X**2<1).set
Interval.open(-1, 1)

>>> where(And(D1<=D2 , D2<3))
Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2))

sympy.stats.variance(X, condition=None, **kwargs)[source]

Variance of a random expression

Expectation of (X-E(X))**2

Examples

>>> from sympy.stats import Die, E, Bernoulli, variance
>>> from sympy import simplify, Symbol

>>> X = Die('X', 6)
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)

>>> variance(2*X)
35/3

>>> simplify(variance(B))
p*(1 - p)

class sympy.stats.Variance[source]

Symbolic expression for the variance.

Examples

>>> from sympy import symbols, Integral
>>> from sympy.stats import Normal, Expectation, Variance, Probability
>>> mu = symbols("mu", positive=True)
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Variance(X)
Variance(X)
>>> Variance(X).evaluate_integral()
sigma**2


Integral representation of the underlying calculations:

>>> Variance(X).rewrite(Integral)
Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))


Integral representation, without expanding the PDF:

>>> Variance(X).rewrite(Probability)
-Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo))


Rewrite the variance in terms of the expectation

>>> Variance(X).rewrite(Expectation)
-Expectation(X)**2 + Expectation(X**2)


Some transformations based on the properties of the variance may happen:

>>> from sympy.abc import a
>>> Y = Normal("Y", 0, 1)
>>> Variance(a*X)
Variance(a*X)


To expand the variance in its expression, use doit():

>>> Variance(a*X).doit()
a**2*Variance(X)
>>> Variance(X + Y)
Variance(X + Y)
>>> Variance(X + Y).doit()
2*Covariance(X, Y) + Variance(X) + Variance(Y)

sympy.stats.covariance(X, Y, condition=None, **kwargs)[source]

Covariance of two random expressions

The expectation that the two variables will rise and fall together

Covariance(X,Y) = E( (X-E(X)) * (Y-E(Y)) )

Examples

>>> from sympy.stats import Exponential, covariance
>>> from sympy import Symbol

>>> rate = Symbol('lambda', positive=True, real=True, finite=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)

>>> covariance(X, X)
lambda**(-2)
>>> covariance(X, Y)
0
>>> covariance(X, Y + rate*X)
1/lambda

class sympy.stats.Covariance[source]

Symbolic expression for the covariance.

Examples

>>> from sympy.stats import Covariance
>>> from sympy.stats import Normal
>>> X = Normal("X", 3, 2)
>>> Y = Normal("Y", 0, 1)
>>> Z = Normal("Z", 0, 1)
>>> W = Normal("W", 0, 1)
>>> cexpr = Covariance(X, Y)
>>> cexpr
Covariance(X, Y)


Evaluate the covariance, $$X$$ and $$Y$$ are independent, therefore zero is the result:

>>> cexpr.evaluate_integral()
0


Rewrite the covariance expression in terms of expectations:

>>> from sympy.stats import Expectation
>>> cexpr.rewrite(Expectation)
Expectation(X*Y) - Expectation(X)*Expectation(Y)


In order to expand the argument, use doit():

>>> from sympy.abc import a, b, c, d
>>> Covariance(a*X + b*Y, c*Z + d*W)
Covariance(a*X + b*Y, c*Z + d*W)
>>> Covariance(a*X + b*Y, c*Z + d*W).doit()
a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y)


This class is aware of some properties of the covariance:

>>> Covariance(X, X).doit()
Variance(X)
>>> Covariance(a*X, b*Y).doit()
a*b*Covariance(X, Y)

sympy.stats.std(X, condition=None, **kwargs)[source]

Standard Deviation of a random expression

Square root of the Expectation of (X-E(X))**2

Examples

>>> from sympy.stats import Bernoulli, std
>>> from sympy import Symbol, simplify

>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)

>>> simplify(std(B))
sqrt(p*(1 - p))

sympy.stats.sample(expr, condition=None, **kwargs)[source]

A realization of the random expression

Examples

>>> from sympy.stats import Die, sample
>>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)

>>> die_roll = sample(X + Y + Z) # A random realization of three dice

sympy.stats.sample_iter(expr, condition=None, numsamples=oo, **kwargs)[source]

Returns an iterator of realizations from the expression given a condition

Parameters

expr: Expr

Random expression to be realized

condition: Expr, optional

A conditional expression

numsamples: integer, optional

Length of the iterator (defaults to infinity)

Examples

>>> from sympy.stats import Normal, sample_iter
>>> X = Normal('X', 0, 1)
>>> expr = X*X + 3
>>> iterator = sample_iter(expr, numsamples=3)
>>> list(iterator)
[12, 4, 7]

sympy.stats.rv.sample_iter_lambdify(expr, condition=None, numsamples=oo, **kwargs)[source]

Uses lambdify for computation. This is fast but does not always work.

sympy.stats.rv.sample_iter_subs(expr, condition=None, numsamples=oo, **kwargs)[source]

Uses subs for computation. This is slow but almost always works.

sympy.stats.rv.sampling_density(expr, given_condition=None, numsamples=1, **kwargs)[source]

Sampling version of density

sympy.stats.rv.sampling_P(condition, given_condition=None, numsamples=1, evalf=True, **kwargs)[source]

Sampling version of P

sympy.stats.rv.sampling_E(expr, given_condition=None, numsamples=1, evalf=True, **kwargs)[source]

Sampling version of E

## Mechanics¶

SymPy Stats employs a relatively complex class hierarchy.

RandomDomains are a mapping of variables to possible values. For example, we might say that the symbol Symbol('x') can take on the values $$\{1,2,3,4,5,6\}$$.

class sympy.stats.rv.RandomDomain[source]

A PSpace, or Probability Space, combines a RandomDomain with a density to provide probabilistic information. For example the above domain could be enhanced by a finite density {1:1/6, 2:1/6, 3:1/6, 4:1/6, 5:1/6, 6:1/6} to fully define the roll of a fair die named x.

class sympy.stats.rv.PSpace[source]

A RandomSymbol represents the PSpace’s symbol ‘x’ inside of SymPy expressions.

class sympy.stats.rv.RandomSymbol[source]

The RandomDomain and PSpace classes are almost never directly instantiated. Instead they are subclassed for a variety of situations.

RandomDomains and PSpaces must be sufficiently general to represent domains and spaces of several variables with arbitrarily complex densities. This generality is often unnecessary. Instead we often build SingleDomains and SinglePSpaces to represent single, univariate events and processes such as a single die or a single normal variable.

class sympy.stats.rv.SinglePSpace[source]
class sympy.stats.rv.SingleDomain[source]

Another common case is to collect together a set of such univariate random variables. A collection of independent SinglePSpaces or SingleDomains can be brought together to form a ProductDomain or ProductPSpace. These objects would be useful in representing three dice rolled together for example.

class sympy.stats.rv.ProductDomain[source]
class sympy.stats.rv.ProductPSpace[source]

The Conditional adjective is added whenever we add a global condition to a RandomDomain or PSpace. A common example would be three independent dice where we know their sum to be greater than 12.

class sympy.stats.rv.ConditionalDomain[source]

We specialize further into Finite and Continuous versions of these classes to represent finite (such as dice) and continuous (such as normals) random variables.

class sympy.stats.frv.FiniteDomain[source]
class sympy.stats.frv.FinitePSpace[source]
class sympy.stats.crv.ContinuousDomain[source]
class sympy.stats.crv.ContinuousPSpace[source]

Additionally there are a few specialized classes that implement certain common random variable types. There is for example a DiePSpace that implements SingleFinitePSpace and a NormalPSpace that implements SingleContinuousPSpace.

class sympy.stats.frv_types.DiePSpace
class sympy.stats.crv_types.NormalPSpace

RandomVariables can be extracted from these objects using the PSpace.values method.

As previously mentioned SymPy Stats employs a relatively complex class structure. Inheritance is widely used in the implementation of end-level classes. This tactic was chosen to balance between the need to allow SymPy to represent arbitrarily defined random variables and optimizing for common cases. This complicates the code but is structured to only be important to those working on extending SymPy Stats to other random variable types.

Users will not use this class structure. Instead these mechanics are exposed through variable creation functions Die, Coin, FiniteRV, Normal, Exponential, etc…. These build the appropriate SinglePSpaces and return the corresponding RandomVariable. Conditional and Product spaces are formed in the natural construction of SymPy expressions and the use of interface functions E, Given, Density, etc….

sympy.stats.Die()
sympy.stats.Normal()

There are some additional functions that may be useful. They are largely used internally.

sympy.stats.rv.random_symbols(expr)[source]

Returns all RandomSymbols within a SymPy Expression.

sympy.stats.rv.pspace(expr)[source]

Returns the underlying Probability Space of a random expression.

For internal use.

Examples

>>> from sympy.stats import pspace, Normal
>>> from sympy.stats.rv import IndependentProductPSpace
>>> X = Normal('X', 0, 1)
>>> pspace(2*X + 1) == X.pspace
True

sympy.stats.rv.rs_swap(a, b)[source]

Build a dictionary to swap RandomSymbols based on their underlying symbol.

i.e. if X = ('x', pspace1) and Y = ('x', pspace2) then X and Y match and the key, value pair {X:Y}` will appear in the result

Inputs: collections a and b of random variables which share common symbols Output: dict mapping RVs in a to RVs in b