scipy.stats.f#
- scipy.stats.f = <scipy.stats._continuous_distns.f_gen object>[source]#
An F continuous random variable.
For the noncentral F distribution, see
ncf
.As an instance of the
rv_continuous
class,f
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.See also
Notes
The F distribution with \(df_1 > 0\) and \(df_2 > 0\) degrees of freedom is the distribution of the ratio of two independent chi-squared distributions with \(df_1\) and \(df_2\) degrees of freedom, after rescaling by \(df_2 / df_1\).
The probability density function for
f
is:\[f(x, df_1, df_2) = \frac{df_2^{df_2/2} df_1^{df_1/2} x^{df_1 / 2-1}} {(df_2+df_1 x)^{(df_1+df_2)/2} B(df_1/2, df_2/2)}\]for \(x > 0\).
f
accepts shape parametersdfn
anddfd
for \(df_1\), the degrees of freedom of the chi-squared distribution in the numerator, and \(df_2\), the degrees of freedom of the chi-squared distribution in the denominator, respectively.The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the
loc
andscale
parameters. Specifically,f.pdf(x, dfn, dfd, loc, scale)
is identically equivalent tof.pdf(y, dfn, dfd) / scale
withy = (x - loc) / scale
. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.Examples
>>> import numpy as np >>> from scipy.stats import f >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> dfn, dfd = 29, 18 >>> mean, var, skew, kurt = f.stats(dfn, dfd, moments='mvsk')
Display the probability density function (
pdf
):>>> x = np.linspace(f.ppf(0.01, dfn, dfd), ... f.ppf(0.99, dfn, dfd), 100) >>> ax.plot(x, f.pdf(x, dfn, dfd), ... 'r-', lw=5, alpha=0.6, label='f pdf')
Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.
Freeze the distribution and display the frozen
pdf
:>>> rv = f(dfn, dfd) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of
cdf
andppf
:>>> vals = f.ppf([0.001, 0.5, 0.999], dfn, dfd) >>> np.allclose([0.001, 0.5, 0.999], f.cdf(vals, dfn, dfd)) True
Generate random numbers:
>>> r = f.rvs(dfn, dfd, size=1000)
And compare the histogram:
>>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> ax.legend(loc='best', frameon=False) >>> plt.show()
Methods
rvs(dfn, dfd, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, dfn, dfd, loc=0, scale=1)
Probability density function.
logpdf(x, dfn, dfd, loc=0, scale=1)
Log of the probability density function.
cdf(x, dfn, dfd, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, dfn, dfd, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, dfn, dfd, loc=0, scale=1)
Survival function (also defined as
1 - cdf
, but sf is sometimes more accurate).logsf(x, dfn, dfd, loc=0, scale=1)
Log of the survival function.
ppf(q, dfn, dfd, loc=0, scale=1)
Percent point function (inverse of
cdf
— percentiles).isf(q, dfn, dfd, loc=0, scale=1)
Inverse survival function (inverse of
sf
).moment(order, dfn, dfd, loc=0, scale=1)
Non-central moment of the specified order.
stats(dfn, dfd, loc=0, scale=1, moments=’mv’)
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(dfn, dfd, loc=0, scale=1)
(Differential) entropy of the RV.
fit(data)
Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(dfn, dfd), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)
Expected value of a function (of one argument) with respect to the distribution.
median(dfn, dfd, loc=0, scale=1)
Median of the distribution.
mean(dfn, dfd, loc=0, scale=1)
Mean of the distribution.
var(dfn, dfd, loc=0, scale=1)
Variance of the distribution.
std(dfn, dfd, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, dfn, dfd, loc=0, scale=1)
Confidence interval with equal areas around the median.