n次伯努利实验，样本相互独立，单次成功概率为p，服从参数为n和p的二项分布:
$$P\{ x= m\} =C_{n}^{m}p^{m}\left( 1-p\right) ^{n-m} \ \ (其中，0<p<1, m=0,1,...,n)$$
累计概率分布函数：
$$F\left( m\right) =P\{ X \leq m\} =\sum ^{m}_{i=0}C_{n}^{i}p^{i}\left( 1-p\right) ^{n-i}$$
二项分布的两种逼近：泊松分布 和 标准正态分布(拉普拉斯中心极限定理)
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当n很大，p较小（稀有事件，一般小于0.1），即np=\(\lambda\)较小，近似逼近泊松分布
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当n很大，p较大，即np也很大，近似逼近标准正态分布 \(Z=\dfrac{X-np}{\sqrt{np\left( 1-p\right) }}\) ，\(X=\sum ^{n}_{i=0}x_{i}\) 对于二项分布，\(x_{i}\)为所有事件和，即成功次数。
abnormality = scipy.stats.binom(total / 100, p).cdf((total - loss) / 100)
abnormality = ((total - loss) - total * p) / math.sqrt(total * p * (1 - p))
在计算二项分布的分布函数\(F_{m}\)时，由于需要多次计算\(C_{n}^{i}\)，对cpu计算资源消耗过大，故采用拉普拉斯中心极限定理或泊分布近似计算。
from scipy import stats
stats.binom?
[0;31mSignature:[0m [0mstats[0m[0;34m.[0m[0mbinom[0m[0;34m([0m[0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwds[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
[0;31mType:[0m binom_gen
[0;31mString form:[0m <scipy.stats._discrete_distns.binom_gen object at 0x7fc9b026c8b0>
[0;31mFile:[0m /opt/conda/envs/blog/lib/python3.8/site-packages/scipy/stats/_discrete_distns.py
[0;31mDocstring:[0m
A binomial discrete random variable.
As an instance of the `rv_discrete` class, `binom` object inherits from it
a collection of generic methods (see below for the full list),
and completes them with details specific for this particular distribution.
Methods
-------
rvs(n, p, loc=0, size=1, random_state=None)
Random variates.
pmf(k, n, p, loc=0)
Probability mass function.
logpmf(k, n, p, loc=0)
Log of the probability mass function.
cdf(k, n, p, loc=0)
Cumulative distribution function.
logcdf(k, n, p, loc=0)
Log of the cumulative distribution function.
sf(k, n, p, loc=0)
Survival function (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(k, n, p, loc=0)
Log of the survival function.
ppf(q, n, p, loc=0)
Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, n, p, loc=0)
Inverse survival function (inverse of ``sf``).
stats(n, p, loc=0, moments='mv')
Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(n, p, loc=0)
(Differential) entropy of the RV.
expect(func, args=(n, p), loc=0, lb=None, ub=None, conditional=False)
Expected value of a function (of one argument) with respect to the distribution.
median(n, p, loc=0)
Median of the distribution.
mean(n, p, loc=0)
Mean of the distribution.
var(n, p, loc=0)
Variance of the distribution.
std(n, p, loc=0)
Standard deviation of the distribution.
interval(alpha, n, p, loc=0)
Endpoints of the range that contains alpha percent of the distribution
Notes
-----
The probability mass function for `binom` is:
.. math::
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
for ``k`` in ``{0, 1,..., n}``.
`binom` takes ``n`` and ``p`` as shape parameters.
The probability mass function above is defined in the "standardized" form.
To shift distribution use the ``loc`` parameter.
Specifically, ``binom.pmf(k, n, p, loc)`` is identically
equivalent to ``binom.pmf(k - loc, n, p)``.
Examples
--------
>>> from scipy.stats import binom
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate a few first moments:
>>> n, p = 5, 0.4
>>> mean, var, skew, kurt = binom.stats(n, p, moments='mvsk')
Display the probability mass function (``pmf``):
>>> x = np.arange(binom.ppf(0.01, n, p),
... binom.ppf(0.99, n, p))
>>> ax.plot(x, binom.pmf(x, n, p), 'bo', ms=8, label='binom pmf')
>>> ax.vlines(x, 0, binom.pmf(x, n, p), colors='b', lw=5, alpha=0.5)
Alternatively, the distribution object can be called (as a function)
to fix the shape and location. This returns a "frozen" RV object holding
the given parameters fixed.
Freeze the distribution and display the frozen ``pmf``:
>>> rv = binom(n, p)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
... label='frozen pmf')
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
Check accuracy of ``cdf`` and ``ppf``:
>>> prob = binom.cdf(x, n, p)
>>> np.allclose(x, binom.ppf(prob, n, p))
True
Generate random numbers:
>>> r = binom.rvs(n, p, size=1000)
[0;31mClass docstring:[0m
A binomial discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `binom` is:
.. math::
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
for ``k`` in ``{0, 1,..., n}``.
`binom` takes ``n`` and ``p`` as shape parameters.
%(after_notes)s
%(example)s
[0;31mCall docstring:[0m
Freeze the distribution for the given arguments.
Parameters
----------
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution. Should include all
the non-optional arguments, may include ``loc`` and ``scale``.
Returns
-------
rv_frozen : rv_frozen instance
The frozen distribution.
stats.binom.ppf?
[0;31mSignature:[0m [0mstats[0m[0;34m.[0m[0mbinom[0m[0;34m.[0m[0mppf[0m[0;34m([0m[0mq[0m[0;34m,[0m [0;34m*[0m[0margs[0m[0;34m,[0m [0;34m**[0m[0mkwds[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m
[0;31mDocstring:[0m
Percent point function (inverse of `cdf`) at q of the given RV.
Parameters
----------
q : array_like
Lower tail probability.
arg1, arg2, arg3,... : array_like
The shape parameter(s) for the distribution (see docstring of the
instance object for more information).
loc : array_like, optional
Location parameter (default=0).
Returns
-------
k : array_like
Quantile corresponding to the lower tail probability, q.
[0;31mFile:[0m /opt/conda/envs/blog/lib/python3.8/site-packages/scipy/stats/_distn_infrastructure.py
[0;31mType:[0m method