from sympy.sets import FiniteSet from sympy import (sqrt, log, exp, FallingFactorial, Rational, Eq, Dummy, piecewise_fold, solveset, Integral) from .rv import (probability, expectation, density, where, given, pspace, cdf, PSpace, characteristic_function, sample, sample_iter, random_symbols, independent, dependent, sampling_density, moment_generating_function, quantile, is_random, sample_stochastic_process) __all__ = ['P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf', 'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std', 'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'median', 'independent', 'random_symbols', 'correlation', 'factorial_moment', 'moment', 'cmoment', 'sampling_density', 'moment_generating_function', 'smoment', 'quantile', 'sample_stochastic_process'] def moment(X, n, c=0, condition=None, *, evaluate=True, **kwargs): """ Return the nth moment of a random expression about c. .. math:: moment(X, c, n) = E((X-c)^{n}) Default value of c is 0. Examples ======== >>> from sympy.stats import Die, moment, E >>> X = Die('X', 6) >>> moment(X, 1, 6) -5/2 >>> moment(X, 2) 91/6 >>> moment(X, 1) == E(X) True """ from sympy.stats.symbolic_probability import Moment if evaluate: return Moment(X, n, c, condition).doit() return Moment(X, n, c, condition).rewrite(Integral) def variance(X, condition=None, **kwargs): """ Variance of a random expression. .. math:: variance(X) = E((X-E(X))^{2}) Examples ======== >>> from sympy.stats import Die, 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) """ if is_random(X) and pspace(X) == PSpace(): from sympy.stats.symbolic_probability import Variance return Variance(X, condition) return cmoment(X, 2, condition, **kwargs) def standard_deviation(X, condition=None, **kwargs): r""" Standard Deviation of a random expression .. math:: std(X) = \sqrt(E((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)) """ return sqrt(variance(X, condition, **kwargs)) std = standard_deviation def entropy(expr, condition=None, **kwargs): """ 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 ========== .. [1] https://en.wikipedia.org/wiki/Entropy_(information_theory) .. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf .. [3] http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf """ pdf = density(expr, condition, **kwargs) base = kwargs.get('b', exp(1)) if isinstance(pdf, dict): return sum([-prob*log(prob, base) for prob in pdf.values()]) return expectation(-log(pdf(expr), base)) def covariance(X, Y, condition=None, **kwargs): """ Covariance of two random expressions. Explanation =========== The expectation that the two variables will rise and fall together .. math:: 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 """ if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y) and pspace(Y) == PSpace()): from sympy.stats.symbolic_probability import Covariance return Covariance(X, Y, condition) return expectation( (X - expectation(X, condition, **kwargs)) * (Y - expectation(Y, condition, **kwargs)), condition, **kwargs) def correlation(X, Y, condition=None, **kwargs): r""" Correlation of two random expressions, also known as correlation coefficient or Pearson's correlation. Explanation =========== The normalized expectation that the two variables will rise and fall together .. math:: correlation(X,Y) = E((X-E(X))(Y-E(Y)) / (\sigma_x \sigma_y)) Examples ======== >>> from sympy.stats import Exponential, correlation >>> from sympy import Symbol >>> rate = Symbol('lambda', positive=True, real=True, finite=True) >>> X = Exponential('X', rate) >>> Y = Exponential('Y', rate) >>> correlation(X, X) 1 >>> correlation(X, Y) 0 >>> correlation(X, Y + rate*X) 1/sqrt(1 + lambda**(-2)) """ return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs) * std(Y, condition, **kwargs)) def cmoment(X, n, condition=None, *, evaluate=True, **kwargs): """ Return the nth central moment of a random expression about its mean. .. math:: cmoment(X, n) = E((X - E(X))^{n}) Examples ======== >>> from sympy.stats import Die, cmoment, variance >>> X = Die('X', 6) >>> cmoment(X, 3) 0 >>> cmoment(X, 2) 35/12 >>> cmoment(X, 2) == variance(X) True """ from sympy.stats.symbolic_probability import CentralMoment if evaluate: return CentralMoment(X, n, condition).doit() return CentralMoment(X, n, condition).rewrite(Integral) def smoment(X, n, condition=None, **kwargs): r""" Return the nth Standardized moment of a random expression. .. math:: smoment(X, n) = E(((X - \mu)/\sigma_X)^{n}) Examples ======== >>> from sympy.stats import skewness, Exponential, smoment >>> from sympy import Symbol >>> rate = Symbol('lambda', positive=True, real=True, finite=True) >>> Y = Exponential('Y', rate) >>> smoment(Y, 4) 9 >>> smoment(Y, 4) == smoment(3*Y, 4) True >>> smoment(Y, 3) == skewness(Y) True """ sigma = std(X, condition, **kwargs) return (1/sigma)**n*cmoment(X, n, condition, **kwargs) def skewness(X, condition=None, **kwargs): r""" Measure of the asymmetry of the probability distribution. Explanation =========== Positive skew indicates that most of the values lie to the right of the mean. .. math:: skewness(X) = E(((X - E(X))/\sigma_X)^{3}) Parameters ========== condition : Expr containing RandomSymbols A conditional expression. skewness(X, X>0) is skewness of X given X > 0 Examples ======== >>> from sympy.stats import skewness, Exponential, Normal >>> from sympy import Symbol >>> X = Normal('X', 0, 1) >>> skewness(X) 0 >>> skewness(X, X > 0) # find skewness given X > 0 (-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2) >>> rate = Symbol('lambda', positive=True, real=True, finite=True) >>> Y = Exponential('Y', rate) >>> skewness(Y) 2 """ return smoment(X, 3, condition=condition, **kwargs) def kurtosis(X, condition=None, **kwargs): r""" Characterizes the tails/outliers of a probability distribution. Explanation =========== Kurtosis of any univariate normal distribution is 3. Kurtosis less than 3 means that the distribution produces fewer and less extreme outliers than the normal distribution. .. math:: kurtosis(X) = E(((X - E(X))/\sigma_X)^{4}) Parameters ========== condition : Expr containing RandomSymbols A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0 Examples ======== >>> from sympy.stats import kurtosis, Exponential, Normal >>> from sympy import Symbol >>> X = Normal('X', 0, 1) >>> kurtosis(X) 3 >>> kurtosis(X, X > 0) # find kurtosis given X > 0 (-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2 >>> rate = Symbol('lamda', positive=True, real=True, finite=True) >>> Y = Exponential('Y', rate) >>> kurtosis(Y) 9 References ========== .. [1] https://en.wikipedia.org/wiki/Kurtosis .. [2] http://mathworld.wolfram.com/Kurtosis.html """ return smoment(X, 4, condition=condition, **kwargs) def factorial_moment(X, n, condition=None, **kwargs): """ The factorial moment is a mathematical quantity defined as the expectation or average of the falling factorial of a random variable. .. math:: factorial-moment(X, n) = E(X(X - 1)(X - 2)...(X - n + 1)) Parameters ========== n: A natural number, n-th factorial moment. condition : Expr containing RandomSymbols A conditional expression. Examples ======== >>> from sympy.stats import factorial_moment, Poisson, Binomial >>> from sympy import Symbol, S >>> lamda = Symbol('lamda') >>> X = Poisson('X', lamda) >>> factorial_moment(X, 2) lamda**2 >>> Y = Binomial('Y', 2, S.Half) >>> factorial_moment(Y, 2) 1/2 >>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1 2 References ========== .. [1] https://en.wikipedia.org/wiki/Factorial_moment .. [2] http://mathworld.wolfram.com/FactorialMoment.html """ return expectation(FallingFactorial(X, n), condition=condition, **kwargs) def median(X, evaluate=True, **kwargs): r""" Calculuates the median of the probability distribution. Explanation =========== Mathematically, median of Probability distribution is defined as all those values of `m` for which the following condition is satisfied .. math:: P(X\leq m) \geq \frac{1}{2} \text{ and} \text{ } P(X\geq m)\geq \frac{1}{2} Parameters ========== X: The random expression whose median is to be calculated. Returns ======= The FiniteSet or an Interval which contains the median of the random expression. Examples ======== >>> from sympy.stats import Normal, Die, median >>> N = Normal('N', 3, 1) >>> median(N) {3} >>> D = Die('D') >>> median(D) {3, 4} References ========== .. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions """ if not is_random(X): return X from sympy.stats.crv import ContinuousPSpace from sympy.stats.drv import DiscretePSpace from sympy.stats.frv import FinitePSpace if isinstance(pspace(X), FinitePSpace): cdf = pspace(X).compute_cdf(X) result = [] for key, value in cdf.items(): if value>= Rational(1, 2) and (1 - value) + \ pspace(X).probability(Eq(X, key)) >= Rational(1, 2): result.append(key) return FiniteSet(*result) if isinstance(pspace(X), ContinuousPSpace) or isinstance(pspace(X), DiscretePSpace): cdf = pspace(X).compute_cdf(X) x = Dummy('x') result = solveset(piecewise_fold(cdf(x) - Rational(1, 2)), x, pspace(X).set) return result raise NotImplementedError("The median of %s is not implemeted."%str(pspace(X))) def coskewness(X, Y, Z, condition=None, **kwargs): r""" Calculates the co-skewness of three random variables. Explanation =========== Mathematically Coskewness is defined as .. math:: coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}} Parameters ========== X : RandomSymbol Random Variable used to calculate coskewness Y : RandomSymbol Random Variable used to calculate coskewness Z : RandomSymbol Random Variable used to calculate coskewness condition : Expr containing RandomSymbols A conditional expression Examples ======== >>> from sympy.stats import coskewness, Exponential, skewness >>> from sympy import symbols >>> p = symbols('p', positive=True) >>> X = Exponential('X', p) >>> Y = Exponential('Y', 2*p) >>> coskewness(X, Y, Y) 0 >>> coskewness(X, Y + X, Y + 2*X) 16*sqrt(85)/85 >>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3) 9*sqrt(170)/85 >>> coskewness(Y, Y, Y) == skewness(Y) True >>> coskewness(X, Y + p*X, Y + 2*p*X) 4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2))) Returns ======= coskewness : The coskewness of the three random variables References ========== .. [1] https://en.wikipedia.org/wiki/Coskewness """ num = expectation((X - expectation(X, condition, **kwargs)) \ * (Y - expectation(Y, condition, **kwargs)) \ * (Z - expectation(Z, condition, **kwargs)), condition, **kwargs) den = std(X, condition, **kwargs) * std(Y, condition, **kwargs) \ * std(Z, condition, **kwargs) return num/den P = probability E = expectation H = entropy