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Classe « Generator »

Méthode numpy.random.Generator.standard_t

Signature de la méthode standard_t

def standard_t(self, df, size=None) 

Description

help(Generator.standard_t)

        standard_t(df, size=None)

        Draw samples from a standard Student's t distribution with `df` degrees
        of freedom.

        A special case of the hyperbolic distribution.  As `df` gets
        large, the result resembles that of the standard normal
        distribution (`standard_normal`).

        Parameters
        ----------
        df : float or array_like of floats
            Degrees of freedom, must be > 0.
        size : int or tuple of ints, optional
            Output shape.  If the given shape is, e.g., ``(m, n, k)``, then
            ``m * n * k`` samples are drawn.  If size is ``None`` (default),
            a single value is returned if ``df`` is a scalar.  Otherwise,
            ``np.array(df).size`` samples are drawn.

        Returns
        -------
        out : ndarray or scalar
            Drawn samples from the parameterized standard Student's t distribution.

        Notes
        -----
        The probability density function for the t distribution is

        .. math:: P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
                  \Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}

        The t test is based on an assumption that the data come from a
        Normal distribution. The t test provides a way to test whether
        the sample mean (that is the mean calculated from the data) is
        a good estimate of the true mean.

        The derivation of the t-distribution was first published in
        1908 by William Gosset while working for the Guinness Brewery
        in Dublin. Due to proprietary issues, he had to publish under
        a pseudonym, and so he used the name Student.

        References
        ----------
        .. [1] Dalgaard, Peter, "Introductory Statistics With R",
               Springer, 2002.
        .. [2] Wikipedia, "Student's t-distribution"
               https://en.wikipedia.org/wiki/Student's_t-distribution

        Examples
        --------
        From Dalgaard page 83 [1]_, suppose the daily energy intake for 11
        women in kilojoules (kJ) is:

        >>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
        ...                    7515, 8230, 8770])

        Does their energy intake deviate systematically from the recommended
        value of 7725 kJ? Our null hypothesis will be the absence of deviation,
        and the alternate hypothesis will be the presence of an effect that could be
        either positive or negative, hence making our test 2-tailed. 

        Because we are estimating the mean and we have N=11 values in our sample,
        we have N-1=10 degrees of freedom. We set our significance level to 95% and 
        compute the t statistic using the empirical mean and empirical standard 
        deviation of our intake. We use a ddof of 1 to base the computation of our 
        empirical standard deviation on an unbiased estimate of the variance (note:
        the final estimate is not unbiased due to the concave nature of the square 
        root).

        >>> np.mean(intake)
        6753.636363636364
        >>> intake.std(ddof=1)
        1142.1232221373727
        >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
        >>> t
        -2.8207540608310198

        We draw 1000000 samples from Student's t distribution with the adequate
        degrees of freedom.

        >>> import matplotlib.pyplot as plt
        >>> rng = np.random.default_rng()
        >>> s = rng.standard_t(10, size=1000000)
        >>> h = plt.hist(s, bins=100, density=True)

        Does our t statistic land in one of the two critical regions found at 
        both tails of the distribution?

        >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))
        0.018318  #random < 0.05, statistic is in critical region

        The probability value for this 2-tailed test is about 1.83%, which is 
        lower than the 5% pre-determined significance threshold. 

        Therefore, the probability of observing values as extreme as our intake
        conditionally on the null hypothesis being true is too low, and we reject 
        the null hypothesis of no deviation. 

        

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