scipy.special.stdtr#
- scipy.special.stdtr(df, t, out=None) = <ufunc 'stdtr'>#
Student t distribution cumulative distribution function
Returns the integral:
\[\frac{\Gamma((df+1)/2)}{\sqrt{\pi df} \Gamma(df/2)} \int_{-\infty}^t (1+x^2/df)^{-(df+1)/2}\, dx\]- Parameters:
- dfarray_like
Degrees of freedom
- tarray_like
Upper bound of the integral
- outndarray, optional
Optional output array for the function results
- Returns:
- scalar or ndarray
Value of the Student t CDF at t
See also
stdtridf
inverse of stdtr with respect to df
stdtrit
inverse of stdtr with respect to t
scipy.stats.t
student t distribution
Notes
The student t distribution is also available as
scipy.stats.t
. Callingstdtr
directly can improve performance compared to thecdf
method ofscipy.stats.t
(see last example below).Examples
Calculate the function for
df=3
att=1
.>>> import numpy as np >>> from scipy.special import stdtr >>> import matplotlib.pyplot as plt >>> stdtr(3, 1) 0.8044988905221148
Plot the function for three different degrees of freedom.
>>> x = np.linspace(-10, 10, 1000) >>> fig, ax = plt.subplots() >>> parameters = [(1, "solid"), (3, "dashed"), (10, "dotted")] >>> for (df, linestyle) in parameters: ... ax.plot(x, stdtr(df, x), ls=linestyle, label=f"$df={df}$") >>> ax.legend() >>> ax.set_title("Student t distribution cumulative distribution function") >>> plt.show()
The function can be computed for several degrees of freedom at the same time by providing a NumPy array or list for df:
>>> stdtr([1, 2, 3], 1) array([0.75 , 0.78867513, 0.80449889])
It is possible to calculate the function at several points for several different degrees of freedom simultaneously by providing arrays for df and t with shapes compatible for broadcasting. Compute
stdtr
at 4 points for 3 degrees of freedom resulting in an array of shape 3x4.>>> dfs = np.array([[1], [2], [3]]) >>> t = np.array([2, 4, 6, 8]) >>> dfs.shape, t.shape ((3, 1), (4,))
>>> stdtr(dfs, t) array([[0.85241638, 0.92202087, 0.94743154, 0.96041658], [0.90824829, 0.97140452, 0.98666426, 0.99236596], [0.93033702, 0.98599577, 0.99536364, 0.99796171]])
The t distribution is also available as
scipy.stats.t
. Callingstdtr
directly can be much faster than calling thecdf
method ofscipy.stats.t
. To get the same results, one must use the following parametrization:scipy.stats.t(df).cdf(x) = stdtr(df, x)
.>>> from scipy.stats import t >>> df, x = 3, 1 >>> stdtr_result = stdtr(df, x) # this can be faster than below >>> stats_result = t(df).cdf(x) >>> stats_result == stdtr_result # test that results are equal True