def test_moment_generating_function(): X = Normal('X', 0, 1) Y = DiscreteUniform('Y', [1, 2, 7]) Z = Poisson('Z', 2) t = symbols('_t') P = Lambda(t, exp(t**2 / 2)) Q = Lambda(t, (exp(7 * t) / 3 + exp(2 * t) / 3 + exp(t) / 3)) R = Lambda(t, exp(2 * exp(t) - 2)) assert moment_generating_function(X).dummy_eq(P) assert moment_generating_function(Y).dummy_eq(Q) assert moment_generating_function(Z).dummy_eq(R)
def test_quadratic_u(): a = Symbol("a", real=True) b = Symbol("b", real=True) X = QuadraticU("x", a, b) Y = QuadraticU("x", 1, 2) # Tests _moment_generating_function assert moment_generating_function(Y)(1) == -15*exp(2) + 27*exp(1) assert moment_generating_function(Y)(2) == -9*exp(4)/2 + 21*exp(2)/2 assert density(X)(x) == (Piecewise((12*(x - a/2 - b/2)**2/(-a + b)**3, And(x <= b, a <= x)), (0, True)))
def test_moment_generating_function(): X = Normal("X", 0, 1) Y = DiscreteUniform("Y", [1, 2, 7]) Z = Poisson("Z", 2) t = symbols("_t") P = Lambda(t, exp(t ** 2 / 2)) Q = Lambda(t, (exp(7 * t) / 3 + exp(2 * t) / 3 + exp(t) / 3)) R = Lambda(t, exp(2 * exp(t) - 2)) assert moment_generating_function(X) == P assert moment_generating_function(Y) == Q assert moment_generating_function(Z) == R
def test_chi(): from sympy import I k = Symbol("k", integer=True) X = Chi('x', k) assert density(X)(x) == 2**(-k / 2 + 1) * x**(k - 1) * exp( -x**2 / 2) / gamma(k / 2) # Tests the characteristic function assert characteristic_function(X)( x) == sqrt(2) * I * x * gamma(k / 2 + S(1) / 2) * hyper( (k / 2 + S(1) / 2, ), (S(3) / 2, ), -x**2 / 2) / gamma(k / 2) + hyper( (k / 2, ), (S(1) / 2, ), -x**2 / 2) # Tests the moment generating function assert moment_generating_function(X)( x) == sqrt(2) * x * gamma(k / 2 + S(1) / 2) * hyper( (k / 2 + S(1) / 2, ), (S(3) / 2, ), x**2 / 2) / gamma(k / 2) + hyper( (k / 2, ), (S(1) / 2, ), x**2 / 2) k = Symbol("k", integer=True, positive=False) raises(ValueError, lambda: Chi('x', k)) k = Symbol("k", integer=False, positive=True) raises(ValueError, lambda: Chi('x', k))
def test_discreteuniform(): # Symbolic a, b, c, t = symbols('a b c t') X = DiscreteUniform('X', [a, b, c]) assert E(X) == (a + b + c) / 3 assert simplify( variance(X) - ((a**2 + b**2 + c**2) / 3 - (a / 3 + b / 3 + c / 3)**2)) == 0 assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S('1/3') Y = DiscreteUniform('Y', range(-5, 5)) # Numeric assert E(Y) == S('-1/2') assert variance(Y) == S('33/4') for x in range(-5, 5): assert P(Eq(Y, x)) == S('1/10') assert P(Y <= x) == S(x + 6) / 10 assert P(Y >= x) == S(5 - x) / 10 assert dict(density(Die('D', 6)).items()) == \ dict(density(DiscreteUniform('U', range(1, 7))).items()) assert characteristic_function(X)( t) == exp(I * a * t) / 3 + exp(I * b * t) / 3 + exp(I * c * t) / 3 assert moment_generating_function(X)( t) == exp(a * t) / 3 + exp(b * t) / 3 + exp(c * t) / 3
def test_bernoulli(): p, a, b, t = symbols("p a b t") X = Bernoulli("B", p, a, b) assert E(X) == a * p + b * (-p + 1) assert density(X)[a] == p assert density(X)[b] == 1 - p assert characteristic_function(X)( t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t) assert moment_generating_function(X)( t) == p * exp(a * t) + (-p + 1) * exp(b * t) X = Bernoulli("B", p, 1, 0) z = Symbol("z") assert E(X) == p assert simplify(variance(X)) == p * (1 - p) assert E(a * X + b) == a * E(X) + b assert simplify(variance(a * X + b)) == simplify(a**2 * variance(X)) assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1)) raises(ValueError, lambda: Bernoulli("B", 1.5)) raises(ValueError, lambda: Bernoulli("B", -0.5)) # issue 8248 assert X.pspace.compute_expectation(1) == 1
def test_binomial_symbolic(): n = 2 p = symbols('p', positive=True) X = Binomial('X', n, p) t = Symbol('t') assert simplify(E(X)) == n * p == simplify(moment(X, 1)) assert simplify(variance(X)) == n * p * (1 - p) == simplify(cmoment(X, 2)) assert cancel((skewness(X) - (1 - 2 * p) / sqrt(n * p * (1 - p)))) == 0 assert cancel((kurtosis(X)) - (3 + (1 - 6 * p * (1 - p)) / (n * p * (1 - p)))) == 0 assert characteristic_function(X)(t) == p**2 * exp( 2 * I * t) + 2 * p * (-p + 1) * exp(I * t) + (-p + 1)**2 assert moment_generating_function(X)( t) == p**2 * exp(2 * t) + 2 * p * (-p + 1) * exp(t) + (-p + 1)**2 # Test ability to change success/failure winnings H, T = symbols('H T') Y = Binomial('Y', n, p, succ=H, fail=T) assert simplify(E(Y) - (n * (H * p + T * (1 - p)))) == 0 # test symbolic dimensions n = symbols('n') B = Binomial('B', n, p) raises(NotImplementedError, lambda: P(B > 2)) assert density(B).dict == Density(BinomialDistribution(n, p, 1, 0)) assert set(density(B).dict.subs(n, 4).doit().keys()) == \ set([S(0), S(1), S(2), S(3), S(4)]) assert set(density(B).dict.subs(n, 4).doit().values()) == \ set([(1 - p)**4, 4*p*(1 - p)**3, 6*p**2*(1 - p)**2, 4*p**3*(1 - p), p**4]) k = Dummy('k', integer=True) assert E(B > 2).dummy_eq( Sum( Piecewise((k * p**k * (1 - p)**(-k + n) * binomial(n, k), (k >= 0) & (k <= n) & (k > 2)), (0, True)), (k, 0, n)))
def test_beta_binomial(): # verify parameters raises(ValueError, lambda: BetaBinomial('b', .2, 1, 2)) raises(ValueError, lambda: BetaBinomial('b', 2, -1, 2)) raises(ValueError, lambda: BetaBinomial('b', 2, 1, -2)) assert BetaBinomial('b', 2, 1, 1) # test numeric values nvals = range(1, 5) alphavals = [S(1) / 4, S.Half, S(3) / 4, 1, 10] betavals = [S(1) / 4, S.Half, S(3) / 4, 1, 10] for n in nvals: for a in alphavals: for b in betavals: X = BetaBinomial('X', n, a, b) assert E(X) == moment(X, 1) assert variance(X) == cmoment(X, 2) # test symbolic n, a, b = symbols('a b n') assert BetaBinomial('x', n, a, b) n = 2 # Because we're using for loops, can't do symbolic n a, b = symbols('a b', positive=True) X = BetaBinomial('X', n, a, b) t = Symbol('t') assert E(X).expand() == moment(X, 1).expand() assert variance(X).expand() == cmoment(X, 2).expand() assert skewness(X) == smoment(X, 3) assert characteristic_function(X)(t) == exp(2*I*t)*beta(a + 2, b)/beta(a, b) +\ 2*exp(I*t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b) assert moment_generating_function(X)(t) == exp(2*t)*beta(a + 2, b)/beta(a, b) +\ 2*exp(t)*beta(a + 1, b + 1)/beta(a, b) + beta(a, b + 2)/beta(a, b)
def test_discreteuniform(): # Symbolic a, b, c, t = symbols('a b c t') X = DiscreteUniform('X', [a, b, c]) assert E(X) == (a + b + c)/3 assert simplify(variance(X) - ((a**2 + b**2 + c**2)/3 - (a/3 + b/3 + c/3)**2)) == 0 assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S('1/3') Y = DiscreteUniform('Y', range(-5, 5)) # Numeric assert E(Y) == S('-1/2') assert variance(Y) == S('33/4') for x in range(-5, 5): assert P(Eq(Y, x)) == S('1/10') assert P(Y <= x) == S(x + 6)/10 assert P(Y >= x) == S(5 - x)/10 assert dict(density(Die('D', 6)).items()) == \ dict(density(DiscreteUniform('U', range(1, 7))).items()) assert characteristic_function(X)(t) == exp(I*a*t)/3 + exp(I*b*t)/3 + exp(I*c*t)/3 assert moment_generating_function(X)(t) == exp(a*t)/3 + exp(b*t)/3 + exp(c*t)/3
def test_discreteuniform(): # Symbolic a, b, c, t = symbols("a b c t") X = DiscreteUniform("X", [a, b, c]) assert E(X) == (a + b + c) / 3 assert (simplify( variance(X) - ((a**2 + b**2 + c**2) / 3 - (a / 3 + b / 3 + c / 3)**2)) == 0) assert P(Eq(X, a)) == P(Eq(X, b)) == P(Eq(X, c)) == S("1/3") Y = DiscreteUniform("Y", range(-5, 5)) # Numeric assert E(Y) == S("-1/2") assert variance(Y) == S("33/4") for x in range(-5, 5): assert P(Eq(Y, x)) == S("1/10") assert P(Y <= x) == S(x + 6) / 10 assert P(Y >= x) == S(5 - x) / 10 assert dict(density(Die("D", 6)).items()) == dict( density(DiscreteUniform("U", range(1, 7))).items()) assert (characteristic_function(X)(t) == exp(I * a * t) / 3 + exp(I * b * t) / 3 + exp(I * c * t) / 3) assert (moment_generating_function(X)(t) == exp(a * t) / 3 + exp(b * t) / 3 + exp(c * t) / 3)
def test_ideal_soliton(): raises(ValueError, lambda: IdealSoliton('sol', -12)) raises(ValueError, lambda: IdealSoliton('sol', 13.2)) raises(ValueError, lambda: IdealSoliton('sol', 0)) f = Function('f') raises(ValueError, lambda: density(IdealSoliton('sol', 10)).pmf(f)) k = Symbol('k', integer=True, positive=True) x = Symbol('x', integer=True, positive=True) t = Symbol('t') sol = IdealSoliton('sol', k) assert density(sol).low == S.One assert density(sol).high == k assert density(sol).dict == Density(density(sol)) assert density(sol).pmf(x) == Piecewise( (1 / k, Eq(x, 1)), (1 / (x * (x - 1)), k >= x), (0, True)) k_vals = [5, 20, 50, 100, 1000] for i in k_vals: assert E(sol.subs(k, i)) == harmonic(i) == moment(sol.subs(k, i), 1) assert variance(sol.subs( k, i)) == (i - 1) + harmonic(i) - harmonic(i)**2 == cmoment( sol.subs(k, i), 2) assert skewness(sol.subs(k, i)) == smoment(sol.subs(k, i), 3) assert kurtosis(sol.subs(k, i)) == smoment(sol.subs(k, i), 4) assert exp(I * t) / 10 + Sum(exp(I * t * x) / (x * x - x), (x, 2, k)).subs( k, 10).doit() == characteristic_function(sol.subs(k, 10))(t) assert exp(t) / 10 + Sum(exp(t * x) / (x * x - x), (x, 2, k)).subs( k, 10).doit() == moment_generating_function(sol.subs(k, 10))(t)
def test_Hermite(): a1 = Symbol("a1", positive=True) a2 = Symbol("a2", negative=True) raises(ValueError, lambda: Hermite("H", a1, a2)) a1 = Symbol("a1", negative=True) a2 = Symbol("a2", positive=True) raises(ValueError, lambda: Hermite("H", a1, a2)) a1 = Symbol("a1", positive=True) x = Symbol("x") H = Hermite("H", a1, a2) assert moment_generating_function(H)(x) == exp( a1 * (exp(x) - 1) + a2 * (exp(2 * x) - 1) ) assert characteristic_function(H)(x) == exp( a1 * (exp(I * x) - 1) + a2 * (exp(2 * I * x) - 1) ) assert E(H) == a1 + 2 * a2 H = Hermite("H", a1=5, a2=4) assert density(H)(2) == 33 * exp(-9) / 2 assert E(H) == 13 assert variance(H) == 21 assert kurtosis(H) == Rational(464, 147) assert skewness(H) == 37 * sqrt(21) / 441
def test_moment_generating_functions(): t = S("t") geometric_mgf = moment_generating_function(Geometric("g", S.Half))(t) assert geometric_mgf.diff(t).subs(t, 0) == 2 logarithmic_mgf = moment_generating_function(Logarithmic("l", S.Half))(t) assert logarithmic_mgf.diff(t).subs(t, 0) == 1 / log(2) negative_binomial_mgf = moment_generating_function( NegativeBinomial("n", 5, Rational(1, 3)) )(t) assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2) poisson_mgf = moment_generating_function(Poisson("p", 5))(t) assert poisson_mgf.diff(t).subs(t, 0) == 5 skellam_mgf = moment_generating_function(Skellam("s", 1, 1))(t) assert skellam_mgf.diff(t).subs(t, 2) == (-exp(-2) + exp(2)) * exp( -2 + exp(-2) + exp(2) ) yule_simon_mgf = moment_generating_function(YuleSimon("y", 3))(t) assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2) zeta_mgf = moment_generating_function(Zeta("z", 5))(t) assert zeta_mgf.diff(t).subs(t, 0) == pi ** 4 / (90 * zeta(5))
def test_moment_generating_functions(): t = S('t') geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t) assert geometric_mgf.diff(t).subs(t, 0) == 2 logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t) assert logarithmic_mgf.diff(t).subs(t, 0) == 1 / log(2) negative_binomial_mgf = moment_generating_function( NegativeBinomial('n', 5, Rational(1, 3)))(t) assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2) poisson_mgf = moment_generating_function(Poisson('p', 5))(t) assert poisson_mgf.diff(t).subs(t, 0) == 5 skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t) assert skellam_mgf.diff(t).subs( t, 2) == (-exp(-2) + exp(2)) * exp(-2 + exp(-2) + exp(2)) yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t) assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2) zeta_mgf = moment_generating_function(Zeta('z', 5))(t) assert zeta_mgf.diff(t).subs(t, 0) == pi**4 / (90 * zeta(5))
def test_dice(): # TODO: Make iid method! X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) a, b, t, p = symbols('a b t p') assert E(X) == 3 + S.Half assert variance(X) == S(35) / 12 assert E(X + Y) == 7 assert E(X + X) == 7 assert E(a * X + b) == a * E(X) + b assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2) assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2) assert cmoment(X, 0) == 1 assert cmoment(4 * X, 3) == 64 * cmoment(X, 3) assert covariance(X, Y) == S.Zero assert covariance(X, X + Y) == variance(X) assert density(Eq(cos(X * S.Pi), 1))[True] == S.Half assert correlation(X, Y) == 0 assert correlation(X, Y) == correlation(Y, X) assert smoment(X + Y, 3) == skewness(X + Y) assert smoment(X, 0) == 1 assert P(X > 3) == S.Half assert P(2 * X > 6) == S.Half assert P(X > Y) == S(5) / 12 assert P(Eq(X, Y)) == P(Eq(X, 1)) assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3) assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3) assert E(X + Y, Eq(X, Y)) == E(2 * X) assert moment(X, 0) == 1 assert moment(5 * X, 2) == 25 * moment(X, 2) assert quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\ (S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\ (S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One)) assert P(X > 3, X > 3) == S.One assert P(X > Y, Eq(Y, 6)) == S.Zero assert P(Eq(X + Y, 12)) == S.One / 36 assert P(Eq(X + Y, 12), Eq(X, 6)) == S.One / 6 assert density(X + Y) == density(Y + Z) != density(X + X) d = density(2 * X + Y**Z) assert d[S(22)] == S.One / 108 and d[S(4100)] == S.One / 216 and S( 3130) not in d assert pspace(X).domain.as_boolean() == Or( *[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]]) assert where(X > 3).set == FiniteSet(4, 5, 6) assert characteristic_function(X)(t) == exp(6 * I * t) / 6 + exp( 5 * I * t) / 6 + exp(4 * I * t) / 6 + exp(3 * I * t) / 6 + exp( 2 * I * t) / 6 + exp(I * t) / 6 assert moment_generating_function(X)( t) == exp(6 * t) / 6 + exp(5 * t) / 6 + exp(4 * t) / 6 + exp( 3 * t) / 6 + exp(2 * t) / 6 + exp(t) / 6
def test_rademacher(): X = Rademacher('X') t = Symbol('t') assert E(X) == 0 assert variance(X) == 1 assert density(X)[-1] == S.Half assert density(X)[1] == S.Half assert characteristic_function(X)(t) == exp(I*t)/2 + exp(-I*t)/2 assert moment_generating_function(X)(t) == exp(t) / 2 + exp(-t) / 2
def test_rademacher(): X = Rademacher('X') t = Symbol('t') assert E(X) == 0 assert variance(X) == 1 assert density(X)[-1] == S.Half assert density(X)[1] == S.Half assert characteristic_function(X)(t) == exp(I * t) / 2 + exp(-I * t) / 2 assert moment_generating_function(X)(t) == exp(t) / 2 + exp(-t) / 2
def test_triangular(): a = Symbol("a") b = Symbol("b") c = Symbol("c") X = Triangular('x', a, b, c) assert str(density(X)(x)) == ("Piecewise(((-2*a + 2*x)/((-a + b)*(-a + c)), (a <= x) & (c > x)), " "(2/(-a + b), Eq(c, x)), ((2*b - 2*x)/((-a + b)*(b - c)), (b >= x) & (c < x)), (0, True))") #Tests moment_generating_function assert moment_generating_function(X)(x).expand() == \ ((-2*(-a + b)*exp(c*x) + 2*(-a + c)*exp(b*x) + 2*(b - c)*exp(a*x))/(x**2*(-a + b)*(-a + c)*(b - c))).expand()
def test_dice(): # TODO: Make iid method! X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) a, b, t, p = symbols('a b t p') assert E(X) == 3 + S.Half assert variance(X) == S(35)/12 assert E(X + Y) == 7 assert E(X + X) == 7 assert E(a*X + b) == a*E(X) + b assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2) assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2) assert cmoment(X, 0) == 1 assert cmoment(4*X, 3) == 64*cmoment(X, 3) assert covariance(X, Y) == S.Zero assert covariance(X, X + Y) == variance(X) assert density(Eq(cos(X*S.Pi), 1))[True] == S.Half assert correlation(X, Y) == 0 assert correlation(X, Y) == correlation(Y, X) assert smoment(X + Y, 3) == skewness(X + Y) assert smoment(X, 0) == 1 assert P(X > 3) == S.Half assert P(2*X > 6) == S.Half assert P(X > Y) == S(5)/12 assert P(Eq(X, Y)) == P(Eq(X, 1)) assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3) assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3) assert E(X + Y, Eq(X, Y)) == E(2*X) assert moment(X, 0) == 1 assert moment(5*X, 2) == 25*moment(X, 2) assert quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\ (S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\ (S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One)) assert P(X > 3, X > 3) == S.One assert P(X > Y, Eq(Y, 6)) == S.Zero assert P(Eq(X + Y, 12)) == S.One/36 assert P(Eq(X + Y, 12), Eq(X, 6)) == S.One/6 assert density(X + Y) == density(Y + Z) != density(X + X) d = density(2*X + Y**Z) assert d[S(22)] == S.One/108 and d[S(4100)] == S.One/216 and S(3130) not in d assert pspace(X).domain.as_boolean() == Or( *[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]]) assert where(X > 3).set == FiniteSet(4, 5, 6) assert characteristic_function(X)(t) == exp(6*I*t)/6 + exp(5*I*t)/6 + exp(4*I*t)/6 + exp(3*I*t)/6 + exp(2*I*t)/6 + exp(I*t)/6 assert moment_generating_function(X)(t) == exp(6*t)/6 + exp(5*t)/6 + exp(4*t)/6 + exp(3*t)/6 + exp(2*t)/6 + exp(t)/6
def test_bernoulli(): p, a, b, t = symbols('p a b t') X = Bernoulli('B', p, a, b) assert E(X) == a*p + b*(-p + 1) assert density(X)[a] == p assert density(X)[b] == 1 - p assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t) assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t) X = Bernoulli('B', p, 1, 0) assert E(X) == p assert simplify(variance(X)) == p*(1 - p) assert E(a*X + b) == a*E(X) + b assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X))
def test_skellam(): mu1 = Symbol('mu1') mu2 = Symbol('mu2') z = Symbol('z') X = Skellam('x', mu1, mu2) assert density(X)(z) == (mu1/mu2)**(z/2) * \ exp(-mu1 - mu2)*besseli(z, 2*sqrt(mu1*mu2)) assert skewness(X).expand() == mu1/(mu1*sqrt(mu1 + mu2) + mu2 * sqrt(mu1 + mu2)) - mu2/(mu1*sqrt(mu1 + mu2) + mu2*sqrt(mu1 + mu2)) assert variance(X).expand() == mu1 + mu2 assert E(X) == mu1 - mu2 assert characteristic_function(X)(z) == exp( mu1*exp(I*z) - mu1 - mu2 + mu2*exp(-I*z)) assert moment_generating_function(X)(z) == exp( mu1*exp(z) - mu1 - mu2 + mu2*exp(-z))
def test_binomial_symbolic(): n = 2 # Because we're using for loops, can't do symbolic n p = symbols('p', positive=True) X = Binomial('X', n, p) t = Symbol('t') assert simplify(E(X)) == n*p == simplify(moment(X, 1)) assert simplify(variance(X)) == n*p*(1 - p) == simplify(cmoment(X, 2)) assert cancel((skewness(X) - (1 - 2*p)/sqrt(n*p*(1 - p)))) == 0 assert characteristic_function(X)(t) == p ** 2 * exp(2 * I * t) + 2 * p * (-p + 1) * exp(I * t) + (-p + 1) ** 2 assert moment_generating_function(X)(t) == p ** 2 * exp(2 * t) + 2 * p * (-p + 1) * exp(t) + (-p + 1) ** 2 # Test ability to change success/failure winnings H, T = symbols('H T') Y = Binomial('Y', n, p, succ=H, fail=T) assert simplify(E(Y) - (n*(H*p + T*(1 - p)))) == 0
def test_uniformsum(): n = Symbol("n", integer=True) _k = Dummy("k") x = Symbol("x") X = UniformSum('x', n) res = Sum((-1)**_k*(-_k + x)**(n - 1)*binomial(n, _k), (_k, 0, floor(x)))/factorial(n - 1) assert density(X)(x).dummy_eq(res) #Tests set functions assert X.pspace.domain.set == Interval(0, n) #Tests the characteristic_function assert characteristic_function(X)(x) == (-I*(exp(I*x) - 1)/x)**n #Tests the moment_generating_function assert moment_generating_function(X)(x) == ((exp(x) - 1)/x)**n
def test_Moyal(): mu = Symbol('mu',real=False) sigma = Symbol('sigma', real=True, positive=True) raises(ValueError, lambda: Moyal('M',mu, sigma)) mu = Symbol('mu', real=True) sigma = Symbol('sigma', real=True, negative=True) raises(ValueError, lambda: Moyal('M',mu, sigma)) sigma = Symbol('sigma', real=True, positive=True) M = Moyal('M', mu, sigma) assert density(M)(z) == sqrt(2)*exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma) assert cdf(M)(z).simplify() == 1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2) assert characteristic_function(M)(z) == 2**(-I*sigma*z)*exp(I*mu*z) \ *gamma(-I*sigma*z + Rational(1, 2))/sqrt(pi) assert E(M) == mu + EulerGamma*sigma + sigma*log(2) assert moment_generating_function(M)(z) == 2**(-sigma*z)*exp(mu*z) \ *gamma(-sigma*z + Rational(1, 2))/sqrt(pi)
def test_bernoulli(): p, a, b, t = symbols('p a b t') X = Bernoulli('B', p, a, b) assert E(X) == a*p + b*(-p + 1) assert density(X)[a] == p assert density(X)[b] == 1 - p assert characteristic_function(X)(t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t) assert moment_generating_function(X)(t) == p * exp(a * t) + (-p + 1) * exp(b * t) X = Bernoulli('B', p, 1, 0) z = Symbol("z") assert E(X) == p assert simplify(variance(X)) == p*(1 - p) assert E(a*X + b) == a*E(X) + b assert simplify(variance(a*X + b)) == simplify(a**2 * variance(X)) assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1)) raises(ValueError, lambda: Bernoulli('B', 1.5)) raises(ValueError, lambda: Bernoulli('B', -0.5))
def test_bernoulli(): p, a, b, t = symbols('p a b t') X = Bernoulli('B', p, a, b) assert E(X) == a * p + b * (-p + 1) assert density(X)[a] == p assert density(X)[b] == 1 - p assert characteristic_function(X)( t) == p * exp(I * a * t) + (-p + 1) * exp(I * b * t) assert moment_generating_function(X)( t) == p * exp(a * t) + (-p + 1) * exp(b * t) X = Bernoulli('B', p, 1, 0) z = Symbol("z") assert E(X) == p assert simplify(variance(X)) == p * (1 - p) assert E(a * X + b) == a * E(X) + b assert simplify(variance(a * X + b)) == simplify(a**2 * variance(X)) assert quantile(X)(z) == Piecewise((nan, (z > 1) | (z < 0)), (0, z <= 1 - p), (1, z <= 1)) Y = Bernoulli('Y', Rational(1, 2)) assert median(Y) == FiniteSet(0, 1) Z = Bernoulli('Z', Rational(2, 3)) assert median(Z) == FiniteSet(1) raises(ValueError, lambda: Bernoulli('B', 1.5)) raises(ValueError, lambda: Bernoulli('B', -0.5)) #issue 8248 assert X.pspace.compute_expectation(1) == 1 p = Rational(1, 5) X = Binomial('X', 5, p) Y = Binomial('Y', 7, 2 * p) Z = Binomial('Z', 9, 3 * p) assert coskewness(Y + Z, X + Y, X + Z).simplify() == 0 assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y).simplify() == \ sqrt(1529)*Rational(12, 16819) assert coskewness(Y + 2*X + Z, X + 2*Y + Z, X + 2*Z + Y, X < 2).simplify() \ == -sqrt(357451121)*Rational(2812, 4646864573)
def test_moment_generating_functions(): t = S('t') geometric_mgf = moment_generating_function(Geometric('g', S(1)/2))(t) assert geometric_mgf.diff(t).subs(t, 0) == 2 logarithmic_mgf = moment_generating_function(Logarithmic('l', S(1)/2))(t) assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2) negative_binomial_mgf = moment_generating_function(NegativeBinomial('n', 5, S(1)/3))(t) assert negative_binomial_mgf.diff(t).subs(t, 0) == S(5)/2 poisson_mgf = moment_generating_function(Poisson('p', 5))(t) assert poisson_mgf.diff(t).subs(t, 0) == 5 yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t) assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == S(3)/2 zeta_mgf = moment_generating_function(Zeta('z', 5))(t) assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5))
def test_dice(): # TODO: Make iid method! X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) a, b, t, p = symbols('a b t p') assert E(X) == 3 + S.Half assert variance(X) == S(35) / 12 assert E(X + Y) == 7 assert E(X + X) == 7 assert E(a * X + b) == a * E(X) + b assert variance(X + Y) == variance(X) + variance(Y) == cmoment(X + Y, 2) assert variance(X + X) == 4 * variance(X) == cmoment(X + X, 2) assert cmoment(X, 0) == 1 assert cmoment(4 * X, 3) == 64 * cmoment(X, 3) assert covariance(X, Y) == S.Zero assert covariance(X, X + Y) == variance(X) assert density(Eq(cos(X * S.Pi), 1))[True] == S.Half assert correlation(X, Y) == 0 assert correlation(X, Y) == correlation(Y, X) assert smoment(X + Y, 3) == skewness(X + Y) assert smoment(X + Y, 4) == kurtosis(X + Y) assert smoment(X, 0) == 1 assert P(X > 3) == S.Half assert P(2 * X > 6) == S.Half assert P(X > Y) == S(5) / 12 assert P(Eq(X, Y)) == P(Eq(X, 1)) assert E(X, X > 3) == 5 == moment(X, 1, 0, X > 3) assert E(X, Y > 3) == E(X) == moment(X, 1, 0, Y > 3) assert E(X + Y, Eq(X, Y)) == E(2 * X) assert moment(X, 0) == 1 assert moment(5 * X, 2) == 25 * moment(X, 2) assert quantile(X)(p) == Piecewise((nan, (p > S.One) | (p < S(0))),\ (S.One, p <= S(1)/6), (S(2), p <= S(1)/3), (S(3), p <= S.Half),\ (S(4), p <= S(2)/3), (S(5), p <= S(5)/6), (S(6), p <= S.One)) assert P(X > 3, X > 3) == S.One assert P(X > Y, Eq(Y, 6)) == S.Zero assert P(Eq(X + Y, 12)) == S.One / 36 assert P(Eq(X + Y, 12), Eq(X, 6)) == S.One / 6 assert density(X + Y) == density(Y + Z) != density(X + X) d = density(2 * X + Y**Z) assert d[S(22)] == S.One / 108 and d[S(4100)] == S.One / 216 and S( 3130) not in d assert pspace(X).domain.as_boolean() == Or( *[Eq(X.symbol, i) for i in [1, 2, 3, 4, 5, 6]]) assert where(X > 3).set == FiniteSet(4, 5, 6) assert characteristic_function(X)(t) == exp(6 * I * t) / 6 + exp( 5 * I * t) / 6 + exp(4 * I * t) / 6 + exp(3 * I * t) / 6 + exp( 2 * I * t) / 6 + exp(I * t) / 6 assert moment_generating_function(X)( t) == exp(6 * t) / 6 + exp(5 * t) / 6 + exp(4 * t) / 6 + exp( 3 * t) / 6 + exp(2 * t) / 6 + exp(t) / 6 # Bayes test for die BayesTest(X > 3, X + Y < 5) BayesTest(Eq(X - Y, Z), Z > Y) BayesTest(X > 3, X > 2) # arg test for die raises(ValueError, lambda: Die('X', -1)) # issue 8105: negative sides. raises(ValueError, lambda: Die('X', 0)) raises(ValueError, lambda: Die('X', 1.5)) # issue 8103: non integer sides. # symbolic test for die n, k = symbols('n, k', positive=True) D = Die('D', n) dens = density(D).dict assert dens == Density(DieDistribution(n)) assert set(dens.subs(n, 4).doit().keys()) == set([1, 2, 3, 4]) assert set(dens.subs(n, 4).doit().values()) == set([S(1) / 4]) k = Dummy('k', integer=True) assert E(D).dummy_eq(Sum(Piecewise((k / n, k <= n), (0, True)), (k, 1, n))) assert variance(D).subs(n, 6).doit() == S(35) / 12 ki = Dummy('ki') cumuf = cdf(D)(k) assert cumuf.dummy_eq( Sum(Piecewise((1 / n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, k))) assert cumuf.subs({n: 6, k: 2}).doit() == S(1) / 3 t = Dummy('t') cf = characteristic_function(D)(t) assert cf.dummy_eq( Sum(Piecewise((exp(ki * I * t) / n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n))) assert cf.subs( n, 3).doit() == exp(3 * I * t) / 3 + exp(2 * I * t) / 3 + exp(I * t) / 3 mgf = moment_generating_function(D)(t) assert mgf.dummy_eq( Sum(Piecewise((exp(ki * t) / n, (ki >= 1) & (ki <= n)), (0, True)), (ki, 1, n))) assert mgf.subs(n, 3).doit() == exp(3 * t) / 3 + exp(2 * t) / 3 + exp(t) / 3
def test_moment_generating_function(): t = symbols('t', positive=True) # Symbolic tests a, b, c = symbols('a b c') mgf = moment_generating_function(Beta('x', a, b))(t) assert mgf == hyper((a, ), (a + b, ), t) mgf = moment_generating_function(Chi('x', a))(t) assert mgf == sqrt(2)*t*gamma(a/2 + S.Half)*\ hyper((a/2 + S.Half,), (Rational(3, 2),), t**2/2)/gamma(a/2) +\ hyper((a/2,), (S.Half,), t**2/2) mgf = moment_generating_function(ChiSquared('x', a))(t) assert mgf == (1 - 2 * t)**(-a / 2) mgf = moment_generating_function(Erlang('x', a, b))(t) assert mgf == (1 - t / b)**(-a) mgf = moment_generating_function(ExGaussian("x", a, b, c))(t) assert mgf == exp(a * t + b**2 * t**2 / 2) / (1 - t / c) mgf = moment_generating_function(Exponential('x', a))(t) assert mgf == a / (a - t) mgf = moment_generating_function(Gamma('x', a, b))(t) assert mgf == (-b * t + 1)**(-a) mgf = moment_generating_function(Gumbel('x', a, b))(t) assert mgf == exp(b * t) * gamma(-a * t + 1) mgf = moment_generating_function(Gompertz('x', a, b))(t) assert mgf == b * exp(b) * expint(t / a, b) mgf = moment_generating_function(Laplace('x', a, b))(t) assert mgf == exp(a * t) / (-b**2 * t**2 + 1) mgf = moment_generating_function(Logistic('x', a, b))(t) assert mgf == exp(a * t) * beta(-b * t + 1, b * t + 1) mgf = moment_generating_function(Normal('x', a, b))(t) assert mgf == exp(a * t + b**2 * t**2 / 2) mgf = moment_generating_function(Pareto('x', a, b))(t) assert mgf == b * (-a * t)**b * uppergamma(-b, -a * t) mgf = moment_generating_function(QuadraticU('x', a, b))(t) assert str(mgf) == ( "(3*(t*(-4*b + (a + b)**2) + 4)*exp(b*t) - " "3*(t*(a**2 + 2*a*(b - 2) + b**2) + 4)*exp(a*t))/(t**2*(a - b)**3)") mgf = moment_generating_function(RaisedCosine('x', a, b))(t) assert mgf == pi**2 * exp(a * t) * sinh(b * t) / (b * t * (b**2 * t**2 + pi**2)) mgf = moment_generating_function(Rayleigh('x', a))(t) assert mgf == sqrt(2)*sqrt(pi)*a*t*(erf(sqrt(2)*a*t/2) + 1)\ *exp(a**2*t**2/2)/2 + 1 mgf = moment_generating_function(Triangular('x', a, b, c))(t) assert str(mgf) == ("(-2*(-a + b)*exp(c*t) + 2*(-a + c)*exp(b*t) + " "2*(b - c)*exp(a*t))/(t**2*(-a + b)*(-a + c)*(b - c))") mgf = moment_generating_function(Uniform('x', a, b))(t) assert mgf == (-exp(a * t) + exp(b * t)) / (t * (-a + b)) mgf = moment_generating_function(UniformSum('x', a))(t) assert mgf == ((exp(t) - 1) / t)**a mgf = moment_generating_function(WignerSemicircle('x', a))(t) assert mgf == 2 * besseli(1, a * t) / (a * t) # Numeric tests mgf = moment_generating_function(Beta('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == hyper((2, ), (3, ), 1) / 2 mgf = moment_generating_function(Chi('x', 1))(t) assert mgf.diff(t).subs(t, 1) == sqrt(2) * hyper( (1, ), (Rational(3, 2), ), S.Half) / sqrt(pi) + hyper( (Rational(3, 2), ), (Rational(3, 2), ), S.Half) + 2 * sqrt(2) * hyper( (2, ), (Rational(5, 2), ), S.Half) / (3 * sqrt(pi)) mgf = moment_generating_function(ChiSquared('x', 1))(t) assert mgf.diff(t).subs(t, 1) == I mgf = moment_generating_function(Erlang('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(ExGaussian("x", 0, 1, 1))(t) assert mgf.diff(t).subs(t, 2) == -exp(2) mgf = moment_generating_function(Exponential('x', 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Gamma('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Gumbel('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == EulerGamma + 1 mgf = moment_generating_function(Gompertz('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == -e * meijerg(((), (1, 1)), ((0, 0, 0), ()), 1) mgf = moment_generating_function(Laplace('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == 1 mgf = moment_generating_function(Logistic('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == beta(1, 1) mgf = moment_generating_function(Normal('x', 0, 1))(t) assert mgf.diff(t).subs(t, 1) == exp(S.Half) mgf = moment_generating_function(Pareto('x', 1, 1))(t) assert mgf.diff(t).subs(t, 0) == expint(1, 0) mgf = moment_generating_function(QuadraticU('x', 1, 2))(t) assert mgf.diff(t).subs(t, 1) == -12 * e - 3 * exp(2) mgf = moment_generating_function(RaisedCosine('x', 1, 1))(t) assert mgf.diff(t).subs(t, 1) == -2*e*pi**2*sinh(1)/\ (1 + pi**2)**2 + e*pi**2*cosh(1)/(1 + pi**2) mgf = moment_generating_function(Rayleigh('x', 1))(t) assert mgf.diff(t).subs(t, 0) == sqrt(2) * sqrt(pi) / 2 mgf = moment_generating_function(Triangular('x', 1, 3, 2))(t) assert mgf.diff(t).subs(t, 1) == -e + exp(3) mgf = moment_generating_function(Uniform('x', 0, 1))(t) assert mgf.diff(t).subs(t, 1) == 1 mgf = moment_generating_function(UniformSum('x', 1))(t) assert mgf.diff(t).subs(t, 1) == 1 mgf = moment_generating_function(WignerSemicircle('x', 1))(t) assert mgf.diff(t).subs(t, 1) == -2*besseli(1, 1) + besseli(2, 1) +\ besseli(0, 1)