1%feature("docstring") OT::KrawtchoukFactory
2"Krawtchouk specific orthonormal univariate polynomial family.
3
4For the :class:`~openturns.Binomial` distribution.
5
6Available constructors:
7    KrawtchoukFactory(*n=1, p=0.5*)
8
9Parameters
10----------
11n : int, :math:`n > 0`
12    Number of experiment parameter of the :class:`~openturns.Binomial`
13    distribution.
14p : float, :math:`0 < p < 1`
15    Success probability parameter of the :class:`~openturns.Binomial`
16    distribution.
17
18Notes
19-----
20Any sequence of orthogonal polynomials has a recurrence formula relating any
21three consecutive polynomials as follows:
22
23.. math::
24
25    P_{i + 1} = (a_i x + b_i) P_i + c_i P_{i - 1}, \quad 1 < i < n
26
27The recurrence coefficients for the Krawtchouk polynomials come analytically
28and read:
29
30.. math::
31
32    \begin{array}{rcl}
33        a_i & = & \displaystyle - \frac{1}
34                                       {\sqrt{(i + 1) (n - i) p (1 - p)}} \\
35        b_i & = & \displaystyle \frac{p (n - i) + i (1 - p)}
36                                     {\sqrt{(i + 1) (n - i) p (1 - p)}} \\
37        c_i & = & \displaystyle - \sqrt{(1 - \frac{1}{i + 1})
38                                        (1 + \frac{1}{n - i})}
39    \end{array}, \quad 1 < i
40
41where :math:`n` and :math:`p` are the parameters of the
42:class:`~openturns.Binomial` distribution.
43
44.. warning::
45
46    The Krawtchouk polynomials are only defined up to a degree :math:`m` equal
47    to :math:`n - 1`. Indeed, for :math:`i = n`, some factors in the
48    denominators of the recurrence coefficients would be equal to zero.
49
50See also
51--------
52StandardDistributionPolynomialFactory
53
54Examples
55--------
56>>> import openturns as ot
57>>> polynomial_factory = ot.KrawtchoukFactory(3, 0.5)
58>>> for i in range(3):
59...     print(polynomial_factory.build(i))
601
61-1.73205 + 1.1547 * X
621.73205 - 3.4641 * X + 1.1547 * X^2"
63
64// ---------------------------------------------------------------------
65
66%feature("docstring") OT::MeixnerFactory::getN
67"Accessor to the number of failures parameter :math:`n`.
68
69Of the :class:`~openturns.Binomial` distribution.
70
71Returns
72-------
73n : int
74    Number of experiments parameter of the :class:`~openturns.Binomial`
75    distribution."
76
77// ---------------------------------------------------------------------
78
79%feature("docstring") OT::MeixnerFactory::getP
80"Accessor to the success probability parameter :math:`p`.
81
82Of the :class:`~openturns.Binomial` distribution.
83
84Returns
85-------
86p : float
87    Success probability parameter of the :class:`~openturns.Binomial`
88    distribution."
89