def test_multiple_normal(): X, Y = Normal('x', 0, 1), Normal('y', 0, 1) p = Symbol("p", positive=True) assert E(X + Y) == 0 assert variance(X + Y) == 2 assert variance(X + X) == 4 assert covariance(X, Y) == 0 assert covariance(2 * X + Y, -X) == -2 * variance(X) assert skewness(X) == 0 assert skewness(X + Y) == 0 assert kurtosis(X) == 3 assert kurtosis(X + Y) == 3 assert correlation(X, Y) == 0 assert correlation(X, X + Y) == correlation(X, X - Y) assert moment(X, 2) == 1 assert cmoment(X, 3) == 0 assert moment(X + Y, 4) == 12 assert cmoment(X, 2) == variance(X) assert smoment(X * X, 2) == 1 assert smoment(X + Y, 3) == skewness(X + Y) assert smoment(X + Y, 4) == kurtosis(X + Y) assert E(X, Eq(X + Y, 0)) == 0 assert variance(X, Eq(X + Y, 0)) == S.Half assert quantile(X)(p) == sqrt(2) * erfinv(2 * p - S.One)
def test_erfcinv(): assert erfcinv(1) is S.Zero assert erfcinv(0) is S.Infinity assert erfcinv(nan) is S.NaN assert erfcinv(x).diff() == -sqrt(pi)*exp(erfcinv(x)**2)/2 raises(ArgumentIndexError, lambda: erfcinv(x).fdiff(2)) assert erfcinv(z).rewrite('erfinv') == erfinv(1-z) assert erfcinv(z).inverse() == erfc
def test_erfi(): assert erfi(nan) is nan assert erfi(oo) is S.Infinity assert erfi(-oo) is S.NegativeInfinity assert erfi(0) is S.Zero assert erfi(I*oo) == I assert erfi(-I*oo) == -I assert erfi(-x) == -erfi(x) assert erfi(I*erfinv(x)) == I*x assert erfi(I*erfcinv(x)) == I*(1 - x) assert erfi(I*erf2inv(0, x)) == I*x assert erfi(I*erf2inv(0, x, evaluate=False)) == I*x # To cover code in erfi assert erfi(I).is_real is False assert erfi(0, evaluate=False).is_real assert erfi(0, evaluate=False).is_zero assert conjugate(erfi(z)) == erfi(conjugate(z)) assert erfi(x).as_leading_term(x) == 2*x/sqrt(pi) assert erfi(x*y).as_leading_term(y) == 2*x*y/sqrt(pi) assert (erfi(x*y)/erfi(y)).as_leading_term(y) == x assert erfi(1/x).as_leading_term(x) == erfi(1/x) assert erfi(z).rewrite('erf') == -I*erf(I*z) assert erfi(z).rewrite('erfc') == I*erfc(I*z) - I assert erfi(z).rewrite('fresnels') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('fresnelc') == (1 - I)*(fresnelc(z*(1 + I)/sqrt(pi)) - I*fresnels(z*(1 + I)/sqrt(pi))) assert erfi(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], z**2)/sqrt(pi) assert erfi(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)/sqrt(pi) assert erfi(z).rewrite('uppergamma') == (sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(S.Pi) - S.One)) assert erfi(z).rewrite('expint') == sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(S.Pi) assert erfi(z).rewrite('tractable') == -I*(-_erfs(I*z)*exp(z**2) + 1) assert expand_func(erfi(I*z)) == I*erf(z) assert erfi(x).as_real_imag() == \ (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2, -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2) assert erfi(x).as_real_imag(deep=False) == \ (erfi(re(x) - I*im(x))/2 + erfi(re(x) + I*im(x))/2, -I*(-erfi(re(x) - I*im(x)) + erfi(re(x) + I*im(x)))/2) assert erfi(w).as_real_imag() == (erfi(w), 0) assert erfi(w).as_real_imag(deep=False) == (erfi(w), 0) raises(ArgumentIndexError, lambda: erfi(x).fdiff(2))
def test_erfc(): assert erfc(nan) is nan assert erfc(oo) is S.Zero assert erfc(-oo) == 2 assert erfc(0) == 1 assert erfc(I*oo) == -oo*I assert erfc(-I*oo) == oo*I assert erfc(-x) == S(2) - erfc(x) assert erfc(erfcinv(x)) == x assert erfc(I).is_real is False assert erfc(0, evaluate=False).is_real assert erfc(0, evaluate=False).is_zero is False assert erfc(erfinv(x)) == 1 - x assert conjugate(erfc(z)) == erfc(conjugate(z)) assert erfc(x).as_leading_term(x) is S.One assert erfc(1/x).as_leading_term(x) == S.Zero assert erfc(z).rewrite('erf') == 1 - erf(z) assert erfc(z).rewrite('erfi') == 1 + I*erfi(I*z) assert erfc(z).rewrite('fresnels') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('fresnelc') == 1 - (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erfc(z).rewrite('hyper') == 1 - 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erfc(z).rewrite('meijerg') == 1 - z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi) assert erfc(z).rewrite('uppergamma') == 1 - sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erfc(z).rewrite('expint') == S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(S.Pi) assert erfc(z).rewrite('tractable') == _erfs(z)*exp(-z**2) assert expand_func(erf(x) + erfc(x)) is S.One assert erfc(x).as_real_imag() == \ (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2, -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2) assert erfc(x).as_real_imag(deep=False) == \ (erfc(re(x) - I*im(x))/2 + erfc(re(x) + I*im(x))/2, -I*(-erfc(re(x) - I*im(x)) + erfc(re(x) + I*im(x)))/2) assert erfc(w).as_real_imag() == (erfc(w), 0) assert erfc(w).as_real_imag(deep=False) == (erfc(w), 0) raises(ArgumentIndexError, lambda: erfc(x).fdiff(2)) assert erfc(x).inverse() == erfcinv
def test_single_normal(): mu = Symbol('mu', real=True) sigma = Symbol('sigma', positive=True) X = Normal('x', 0, 1) Y = X * sigma + mu assert E(Y) == mu assert variance(Y) == sigma**2 pdf = density(Y) x = Symbol('x', real=True) assert (pdf(x) == 2**S.Half * exp(-(x - mu)**2 / (2 * sigma**2)) / (2 * pi**S.Half * sigma)) assert P(X**2 < 1) == erf(2**S.Half / 2) assert quantile(Y)(x) == Intersection( S.Reals, FiniteSet( sqrt(2) * sigma * (sqrt(2) * mu / (2 * sigma) + erfinv(2 * x - 1)))) assert E(X, Eq(X, mu)) == mu
def test_multiple_normal(): X, Y = Normal('x', 0, 1), Normal('y', 0, 1) p = Symbol("p", positive=True) assert E(X + Y) == 0 assert variance(X + Y) == 2 assert variance(X + X) == 4 assert covariance(X, Y) == 0 assert covariance(2*X + Y, -X) == -2*variance(X) assert skewness(X) == 0 assert skewness(X + Y) == 0 assert correlation(X, Y) == 0 assert correlation(X, X + Y) == correlation(X, X - Y) assert moment(X, 2) == 1 assert cmoment(X, 3) == 0 assert moment(X + Y, 4) == 12 assert cmoment(X, 2) == variance(X) assert smoment(X*X, 2) == 1 assert smoment(X + Y, 3) == skewness(X + Y) assert E(X, Eq(X + Y, 0)) == 0 assert variance(X, Eq(X + Y, 0)) == S.Half assert quantile(X)(p) == sqrt(2)*erfinv(2*p - S.One)
def test_erfinv_evalf(): assert abs( erfinv(Float(0.2)) - 0.179143454621292 ) < 1E-13
def test_erf(): assert erf(nan) is nan assert erf(oo) == 1 assert erf(-oo) == -1 assert erf(0) is S.Zero assert erf(I*oo) == oo*I assert erf(-I*oo) == -oo*I assert erf(-2) == -erf(2) assert erf(-x*y) == -erf(x*y) assert erf(-x - y) == -erf(x + y) assert erf(erfinv(x)) == x assert erf(erfcinv(x)) == 1 - x assert erf(erf2inv(0, x)) == x assert erf(erf2inv(0, x, evaluate=False)) == x # To cover code in erf assert erf(erf2inv(0, erf(erfcinv(1 - erf(erfinv(x)))))) == x assert erf(I).is_real is False assert erf(0, evaluate=False).is_real assert erf(0, evaluate=False).is_zero assert conjugate(erf(z)) == erf(conjugate(z)) assert erf(x).as_leading_term(x) == 2*x/sqrt(pi) assert erf(x*y).as_leading_term(y) == 2*x*y/sqrt(pi) assert (erf(x*y)/erf(y)).as_leading_term(y) == x assert erf(1/x).as_leading_term(x) == S.One assert erf(z).rewrite('uppergamma') == sqrt(z**2)*(1 - erfc(sqrt(z**2)))/z assert erf(z).rewrite('erfc') == S.One - erfc(z) assert erf(z).rewrite('erfi') == -I*erfi(I*z) assert erf(z).rewrite('fresnels') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('fresnelc') == (1 + I)*(fresnelc(z*(1 - I)/sqrt(pi)) - I*fresnels(z*(1 - I)/sqrt(pi))) assert erf(z).rewrite('hyper') == 2*z*hyper([S.Half], [3*S.Half], -z**2)/sqrt(pi) assert erf(z).rewrite('meijerg') == z*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)/sqrt(pi) assert erf(z).rewrite('expint') == sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(S.Pi) assert limit(exp(x)*exp(x**2)*(erf(x + 1/exp(x)) - erf(x)), x, oo) == \ 2/sqrt(pi) assert limit((1 - erf(z))*exp(z**2)*z, z, oo) == 1/sqrt(pi) assert limit((1 - erf(x))*exp(x**2)*sqrt(pi)*x, x, oo) == 1 assert limit(((1 - erf(x))*exp(x**2)*sqrt(pi)*x - 1)*2*x**2, x, oo) == -1 assert limit(erf(x)/x, x, 0) == 2/sqrt(pi) assert limit(x**(-4) - sqrt(pi)*erf(x**2) / (2*x**6), x, 0) == S(1)/3 assert erf(x).as_real_imag() == \ (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2, -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2) assert erf(x).as_real_imag(deep=False) == \ (erf(re(x) - I*im(x))/2 + erf(re(x) + I*im(x))/2, -I*(-erf(re(x) - I*im(x)) + erf(re(x) + I*im(x)))/2) assert erf(w).as_real_imag() == (erf(w), 0) assert erf(w).as_real_imag(deep=False) == (erf(w), 0) # issue 13575 assert erf(I).as_real_imag() == (0, -I*erf(I)) raises(ArgumentIndexError, lambda: erf(x).fdiff(2)) assert erf(x).inverse() == erfinv
def test_single_normal(): mu = Symbol('mu', real=True, finite=True) sigma = Symbol('sigma', real=True, positive=True, finite=True) X = Normal('x', 0, 1) Y = X*sigma + mu assert simplify(E(Y)) == mu assert simplify(variance(Y)) == sigma**2 pdf = density(Y) x = Symbol('x') assert (pdf(x) == 2**S.Half*exp(-(mu - x)**2/(2*sigma**2))/(2*pi**S.Half*sigma)) assert P(X**2 < 1) == erf(2**S.Half/2) assert quantile(Y)(x) == Intersection(S.Reals, FiniteSet(sqrt(2)*sigma*(sqrt(2)*mu/(2*sigma) + erfinv(2*x - 1)))) assert E(X, Eq(X, mu)) == mu