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 |
|
Probability |
|
Expected value |
|
Variance |
|
Probability Density Function |
|
Produce a realization |
|
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¶
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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
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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
>>> 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}
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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 {Heads: 1/2, Tails: 1/2}
References
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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}
See also
References
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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
>>> 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}
References
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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
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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
Discrete Types¶
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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
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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
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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
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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
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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
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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
Continuous Types¶
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sympy.stats.Arcsin(name, a=0, b=1)[source]¶ Creates a Continuous Random Variable with Arcsine distribution.
The density of the Arcsine 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
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sympy.stats.Benini(name, alpha, beta, sigma)[source]¶ Creates a Continuous Random Variable with 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 distrubtion 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
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sympy.stats.Beta(name, alpha, beta)[source]¶ Creates a Continuous Random Variable with 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))) #doctest: +SKIP alpha*beta/((alpha + beta)**2*(alpha + beta + 1))
References
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sympy.stats.BetaPrime(name, alpha, beta)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.Cauchy(name, x0, gamma)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.Chi(name, k)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.ChiNoncentral(name, k, l)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.ChiSquared(name, k)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.Dagum(name, p, a, b)[source]¶ Creates a Continuous Random Variable with 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
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sympy.stats.Erlang(name, k, l)[source]¶ Creates a Continuous Random Variable with 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
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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 >>> from sympy import Symbol
>>> l = Symbol("lambda", positive=True) >>> z = Symbol("z")
>>> X = Exponential("x", l)
>>> density(X)(z) lambda*exp(-lambda*z)
>>> cdf(X)(z) Piecewise((1 - exp(-lambda*z), z >= 0), (0, True))
>>> 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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 sqaure matrix,
\(\sigma^2 > 0\) the variance
- Returns
A RandomSymbol.
Examples
>>> from sympy.stats import Normal, density, E, std, cdf, skewness >>> from sympy import Symbol, simplify, pprint, factor, together, factor_terms
>>> mu = Symbol("mu") >>> sigma = Symbol("sigma", positive=True) >>> z = Symbol("z") >>> y = Symbol("y") >>> 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
>>> 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)) /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
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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
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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
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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
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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
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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
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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
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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) # doctest: +SKIP -a/(-a + b) + z/(-a + b)
>>> simplify(E(X)) a/2 + b/2
>>> simplify(variance(X)) a**2/12 - a*b/6 + b**2/12
References
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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
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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
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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
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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
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sympy.stats.ContinuousRV(symbol, density, set=Interval(-oo, oo))[source]¶ Creates 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 needed 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
Interface¶
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sympy.stats.P(condition, given_condition=None, numsamples=None, evaluate=True, **kwargs)¶ 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
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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))
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sympy.stats.E(expr, condition=None, numsamples=None, evaluate=True, **kwargs)¶ 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
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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
Expectationinto its expression, usedoit():>>> 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)
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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))
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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
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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))
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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)
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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)
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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
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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)
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sympy.stats.std(X, condition=None, **kwargs)¶ 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))
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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
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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) # doctest: +SKIP [12, 4, 7]
See also
Sample,sampling_P,sampling_E,sample_iter_lambdify,sample_iter_subs
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 \(\{1,2,3,4,5,6\}\).
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.
A RandomSymbol represents the PSpace’s symbol ‘x’ inside of SymPy expressions.
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.
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.
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.
We specialize further into Finite and Continuous versions of these classes to represent finite (such as dice) and continuous (such as normals) random variables.
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….
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sympy.stats.Die()¶
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sympy.stats.Normal()¶
There are some additional functions that may be useful. They are largely used internally.
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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
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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)andY = ('x', pspace2)thenXandYmatch and the key, value pair{X:Y}will appear in the resultInputs: collections a and b of random variables which share common symbols Output: dict mapping RVs in a to RVs in b
