def __new__(cls, mu, omega, symbol=None):
mu, omega = sympify(mu), sympify(omega)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2 * mu**mu / (gamma(mu) * omega**mu) * x**(2 * mu - 1) * exp(
-mu / omega * x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
return obj
def __new__(cls, nu, symbol=None):
nu = sympify(nu)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1 / (sqrt(nu) * beta_fn(S(1) / 2, nu / 2)) * (1 + x**2 / nu)**(
-(nu + 1) / 2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, a, symbol=None):
a = sympify(a)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = sqrt(2 / pi) * x**2 * exp(-x**2 / (2 * a**2)) / a**3
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
return obj
def __new__(cls, mu, s, symbol=None):
mu, s = sympify(mu), sympify(s)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(x-mu)/s)/(s*(1+exp(-(x-mu)/s))**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, mu, s, symbol=None):
mu, s = sympify(mu), sympify(s)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(x - mu) / s) / (s * (1 + exp(-(x - mu) / s))**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, R, symbol=None):
R = sympify(R)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2 / (pi * R**2) * sqrt(R**2 - x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(-R, R))
return obj
def __new__(cls, alpha, beta, sigma, symbol = None):
alpha, beta, sigma = sympify(alpha), sympify(beta), sympify(sigma)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = (exp(-alpha*log(x/sigma)-beta*log(x/sigma)**2)
*(alpha/x+2*beta*log(x/sigma)/x))
obj = SingleContinuousPSpace.__new__(cls, x, pdf,
set = Interval(sigma, oo))
return obj
def __new__(cls, a, symbol = None):
a = sympify(a)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(0, oo))
return obj
def __new__(cls, rate, symbol=None):
_value_check(rate > 0, "Rate must be positive.")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = rate * exp(-rate*x)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.rate = rate
return obj
def __new__(cls, R, symbol=None):
R = sympify(R)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2/(pi*R**2)*sqrt(R**2-x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(-R, R))
return obj
def __new__(cls, n, symbol=None):
n = sympify(n)
x = symbol or SingleContinuousPSpace.create_symbol()
k = Dummy("k")
pdf =1/factorial(n-1)*Sum((-1)**k*binomial(n,k)*(x-k)**(n-1), (k,0,floor(x)))
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0,n))
return obj
def __new__(cls, mean, std, symbol=None):
mean, std = sympify(mean), sympify(std)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(log(x)-mean)**2 / (2*std**2)) / (x*sqrt(2*pi)*std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.mean = mean
obj.std = std
return obj
def __new__(cls, mean, std, symbol = None):
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(x-mean)**2 / (2*std**2)) / (sqrt(2*pi)*std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.mean = mean
obj.std = std
obj.variance = std**2
return obj
def __new__(cls, mean, std, symbol=None):
_value_check(std > 0, "Standard deviation must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(x-mean)**2 / (2*std**2)) / (sqrt(2*pi)*std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.mean = mean
obj.std = std
obj.variance = std**2
return obj
def __new__(cls, xm, alpha, symbol=None):
_value_check(xm > 0, "Xm must be positive")
_value_check(alpha > 0, "Alpha must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = alpha * xm**alpha / x**(alpha+1)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(xm, oo))
obj.xm = xm
obj.alpha = alpha
return obj
def __new__(cls, alpha, beta, sigma, symbol=None):
alpha, beta, sigma = sympify(alpha), sympify(beta), sympify(sigma)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = (exp(-alpha * log(x / sigma) - beta * log(x / sigma)**2) *
(alpha / x + 2 * beta * log(x / sigma) / x))
obj = SingleContinuousPSpace.__new__(cls,
x,
pdf,
set=Interval(sigma, oo))
return obj
def __new__(cls, n, symbol=None):
n = sympify(n)
x = symbol or SingleContinuousPSpace.create_symbol()
k = Dummy("k")
pdf = 1 / factorial(n - 1) * Sum(
(-1)**k * binomial(n, k) * (x - k)**(n - 1), (k, 0, floor(x)))
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, n))
return obj
def __new__(cls, left, right, symbol=None):
left, right = sympify(left), sympify(right)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = Piecewise((S.Zero, x < left), (S.Zero, x > right),
(S.One / (right - left), True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.left = left
obj.right = right
return obj
def __new__(cls, mean, std, symbol=None):
mean, std = sympify(mean), sympify(std)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(log(x) - mean)**2 /
(2 * std**2)) / (x * sqrt(2 * pi) * std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.mean = mean
obj.std = std
return obj
def __new__(cls, k, theta, symbol=None):
_value_check(k > 0, "k must be positive")
_value_check(theta > 0, "Theta must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = x**(k-1) * exp(-x/theta) / (gamma(k)*theta**k)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.k = k
obj.theta = theta
return obj
def __new__(cls, rate, symbol=None):
rate = sympify(rate)
_value_check(rate > 0, "Rate must be positive.")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = rate * exp(-rate * x)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.rate = rate
return obj
def __new__(cls, left, right, symbol=None):
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = Piecewise(
(S.Zero, x<left),
(S.Zero, x>right),
(S.One/(right-left), True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.left = left
obj.right = right
return obj
def __new__(cls, a, b, c, symbol=None):
a, b, c = sympify(a), sympify(b), sympify(c)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = Piecewise(
(2*(x-a)/((b-a)*(c-a)), And(a<=x, x<c)),
(2/(b-a), Eq(x,c)),
(2*(b-x)/((b-a)*(b-c)), And(c<x, x<=b)),
(S.Zero, True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, a, b, c, symbol=None):
a, b, c = sympify(a), sympify(b), sympify(c)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = Piecewise(
(2 * (x - a) / ((b - a) * (c - a)), And(a <= x, x < c)),
(2 / (b - a), Eq(x, c)),
(2 * (b - x) / ((b - a) * (b - c)), And(c < x, x <= b)),
(S.Zero, True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, xm, alpha, symbol=None):
xm, alpha = sympify(xm), sympify(alpha)
_value_check(xm > 0, "Xm must be positive")
_value_check(alpha > 0, "Alpha must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = alpha * xm**alpha / x**(alpha + 1)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(xm, oo))
obj.xm = xm
obj.alpha = alpha
return obj
def __new__(cls, alpha, beta, symbol=None):
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
alpha, beta = sympify(alpha), sympify(beta)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = beta * (x/alpha)**(beta-1) * exp(-(x/alpha)**beta) / alpha
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.alpha = alpha
obj.beta = beta
return obj
def __new__(cls, mean, std, symbol=None):
mean, std = sympify(mean), sympify(std)
_value_check(std > 0, "Standard deviation must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = exp(-(x - mean)**2 / (2 * std**2)) / (sqrt(2 * pi) * std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.mean = mean
obj.std = std
obj.variance = std**2
return obj
def __new__(cls, alpha, beta, symbol=None):
alpha, beta = sympify(alpha), sympify(beta)
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = x**(alpha - 1) * (1 - x)**(beta - 1) / beta_fn(alpha, beta)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, 1))
obj.alpha = alpha
obj.beta = beta
return obj
def __new__(cls, alpha, beta, symbol=None):
alpha, beta = sympify(alpha), sympify(beta)
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = x**(alpha-1) * (1-x)**(beta-1) / beta_fn(alpha, beta)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, 1))
obj.alpha = alpha
obj.beta = beta
return obj
def __new__(cls, k, theta, symbol=None):
k, theta = sympify(k), sympify(theta)
_value_check(k > 0, "k must be positive")
_value_check(theta > 0, "Theta must be positive")
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = x**(k - 1) * exp(-x / theta) / (gamma(k) * theta**k)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.k = k
obj.theta = theta
return obj
def compute_cdf(self, expr, **kwargs):
from sympy import Lambda, Min
z = Dummy('z', real=True, bounded=True)
result = SingleContinuousPSpace.compute_cdf(self, expr, **kwargs)
result = result(z).subs({Min(z, self.right): z,
Min(z, self.left, self.right): self.left})
return Lambda(z, result)
def __new__(cls, name, mu, omega):
mu, omega = sympify(mu), sympify(omega)
x = Symbol(name)
pdf = 2 * mu**mu / (gamma(mu) * omega**mu) * x**(2 * mu - 1) * exp(
-mu / omega * x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
return obj
def __new__(cls, name, nu):
nu = sympify(nu)
x = Symbol(name)
pdf = 1 / (sqrt(nu) * beta_fn(S(1) / 2, nu / 2)) * (1 + x**2 / nu)**(
-(nu + 1) / 2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def ContinuousRV(symbol, density, set=Interval(-oo, oo)):
"""
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
"""
return SingleContinuousPSpace(symbol, density, set).value
def __new__(cls, name, R):
R = sympify(R)
x = Symbol(name)
pdf = 2/(pi*R**2)*sqrt(R**2-x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(-R, R))
return obj
def __new__(cls, name, R):
R = sympify(R)
x = Symbol(name)
pdf = 2/(pi*R**2)*sqrt(R**2 - x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(-R, R))
return obj
def __new__(cls, name, mu, s):
mu, s = sympify(mu), sympify(s)
x = Symbol(name)
pdf = exp(-(x-mu)/s)/(s*(1+exp(-(x-mu)/s))**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, name, a):
a = sympify(a)
x = Symbol(name)
pdf = sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
return obj
def __new__(cls, name, mu, s):
mu, s = sympify(mu), sympify(s)
x = Symbol(name)
pdf = exp(-(x - mu)/s)/(s*(1 + exp(-(x - mu)/s))**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, name, a):
a = sympify(a)
x = Symbol(name)
pdf = sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(0, oo))
return obj
def integrate(self, expr, rvs=None, **kwargs):
from sympy import Max, Min
result = SingleContinuousPSpace.integrate(self, expr, rvs, **kwargs)
result = result.subs({
Max(self.left, self.right): self.right,
Min(self.left, self.right): self.left
})
return result
def compute_cdf(self, expr, **kwargs):
from sympy import Lambda, Min
z = Dummy('z', real=True, bounded=True)
result = SingleContinuousPSpace.compute_cdf(self, expr, **kwargs)
result = result(z).subs({
Min(z, self.right): z,
Min(z, self.left, self.right): self.left
})
return Lambda(z, result)
def __new__(cls, name, left, right):
left, right = sympify(left), sympify(right)
x = Symbol(name)
pdf = Piecewise((S.One / (right - left), And(left <= x, x <= right)),
(S.Zero, True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.left = left
obj.right = right
return obj
def __new__(cls, name, rate):
rate = sympify(rate)
_value_check(rate > 0, "Rate must be positive.")
x = Symbol(name)
pdf = rate * exp(-rate*x)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.rate = rate
return obj
def __new__(cls, name, left, right):
left, right = sympify(left), sympify(right)
x = Symbol(name)
pdf = Piecewise(
(S.One/(right - left), And(left <= x, x <= right)),
(S.Zero, True))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.left = left
obj.right = right
return obj
def __new__(cls, name, xm, alpha):
xm, alpha = sympify(xm), sympify(alpha)
_value_check(xm > 0, "Xm must be positive")
_value_check(alpha > 0, "Alpha must be positive")
x = Symbol(name)
pdf = alpha * xm**alpha / x**(alpha + 1)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(xm, oo))
obj.xm = xm
obj.alpha = alpha
return obj
def __new__(cls, name, alpha, beta):
alpha, beta = sympify(alpha), sympify(beta)
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
x = Symbol(name)
pdf = beta * (x/alpha)**(beta - 1) * exp(-(x/alpha)**beta) / alpha
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.alpha = alpha
obj.beta = beta
return obj
def __new__(cls, name, mean, std):
mean, std = sympify(mean), sympify(std)
_value_check(std > 0, "Standard deviation must be positive")
x = Symbol(name)
pdf = exp(-(x - mean)**2 / (2*std**2)) / (sqrt(2*pi)*std)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
obj.mean = mean
obj.std = std
obj.variance = std**2
return obj
def __new__(cls, name, k, theta):
k, theta = sympify(k), sympify(theta)
_value_check(k > 0, "k must be positive")
_value_check(theta > 0, "Theta must be positive")
x = Symbol(name)
pdf = x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.k = k
obj.theta = theta
return obj
def __new__(cls, name, alpha, beta):
alpha, beta = sympify(alpha), sympify(beta)
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
x = Symbol(name)
pdf = beta * (x/alpha)**(beta-1) * exp(-(x/alpha)**beta) / alpha
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
obj.alpha = alpha
obj.beta = beta
return obj
def __new__(cls, x0, gamma, symbol=None):
x0, gamma = sympify(x0), sympify(gamma)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1 / (pi * gamma * (1 + ((x - x0) / gamma)**2))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, mu, b, symbol=None):
mu, b = sympify(mu), sympify(b)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1/(2*b)*exp(-Abs(x-mu)/b)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, p, a, b, symbol = None):
p, a, b = sympify(p), sympify(a), sympify(b)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = a*p/x*((x/b)**(a*p)/(((x/b)**a+1)**(p+1)))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, k, symbol = None):
k = sympify(k)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2**(1-k/2)*x**(k-1)*exp(-x**2/2)/gamma(k/2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(0, oo))
return obj
def __new__(cls, x0, gamma, symbol = None):
x0, gamma = sympify(x0), sympify(gamma)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1/(pi*gamma*(1+((x-x0)/gamma)**2))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, mu, b, symbol=None):
mu, b = sympify(mu), sympify(b)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1 / (2 * b) * exp(-Abs(x - mu) / b)
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, p, a, b, symbol=None):
p, a, b = sympify(p), sympify(a), sympify(b)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = a * p / x * ((x / b)**(a * p) / (((x / b)**a + 1)**(p + 1)))
obj = SingleContinuousPSpace.__new__(cls, x, pdf)
return obj
def __new__(cls, k, symbol=None):
k = sympify(k)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2**(1 - k / 2) * x**(k - 1) * exp(-x**2 / 2) / gamma(k / 2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(0, oo))
return obj
def __new__(cls, mu, omega, symbol = None):
mu, omega = sympify(mu), sympify(omega)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 2*mu**mu/(gamma(mu)*omega**mu)*x**(2*mu-1)*exp(-mu/omega*x**2)
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set = Interval(0, oo))
return obj
def __new__(cls, a, b, symbol=None):
a, b = sympify(a), sympify(b)
x = symbol or SingleContinuousPSpace.create_symbol()
pdf = 1 / (pi * sqrt((x - a) * (b - x)))
obj = SingleContinuousPSpace.__new__(cls, x, pdf, set=Interval(a, b))
return obj