Ejemplo n.º 1
0
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)
Ejemplo n.º 2
0
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
Ejemplo n.º 3
0
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
Ejemplo n.º 4
0
def symstat():
    """
    check out https://docs.sympy.org/latest/modules/stats.html
    """
    from sympy.stats import E

    X, Z = Die('X', 6), Die('Z', 6)
    Y = X + Z


    print('E[X]: {}'.format(E(X)))
    print('\n')
    print('E[Y]: {}'.format(E(Y)))
    print('\n')
    for x in list(density(X)):
        print('E[Y|X = {0}] = {1}'.format(x, E(Y, X=x)))  # this conditioning isn't quite right
        print('\n')
    for y in list(density(Y)):
        print('E[X|Y = {0}] = {1}'.format(y, E(X, Y=y)))
        print('\n')
    print('Var(X): {}'.format(variance(X)))
    print('\n')
    print('Var(Y): {}'.format(variance(Y)))
    print('\n')
    print('Cov(X, Y): {}'.format(covariance(X, X + Y)))
    print('\n')
    print('rho(X, Y): {}'.format(correlation(X, X + Y)))
    print('\n')
    print('Cov(X, Z): {}'.format(covariance(X, Y)))
    print('\n')
Ejemplo n.º 5
0
def test_dice():
    # TODO: Make iid method!
    X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
    a, b = symbols('a b')

    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 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)
Ejemplo n.º 6
0
def test_dice():
    # TODO: Make iid method!
    X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6)
    a, b = symbols('a b')

    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 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)
Ejemplo n.º 7
0
def test_multiple_normal():
    X, Y = Normal('x', 0, 1), Normal('y', 0, 1)

    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
Ejemplo n.º 8
0
def test_multiple_normal():
    X, Y = Normal('x', 0, 1), Normal('y', 0, 1)

    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
Ejemplo n.º 9
0
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)
Ejemplo n.º 10
0
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