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# scipy.stats.ncf

`scipy.stats.``ncf` = <scipy.stats._continuous_distns.ncf_gen object at 0x4508356c>

  A non-central F distribution continuous random variable.

  As an instance of the `rv_continuous` class, `ncf` 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.

  Notes

  The probability density function for `ncf` is:

      ncf.pdf(x, df1, df2, nc) = exp(nc/2 + nc*df1*x/(2*(df1*x+df2))) *
                  df1**(df1/2) * df2**(df2/2) * x**(df1/2-1) *
                  (df2+df1*x)**(-(df1+df2)/2) *
                  gamma(df1/2)*gamma(1+df2/2) *
                  L^{v1/2-1}^{v2/2}(-nc*v1*x/(2*(v1*x+v2))) /
                  (B(v1/2, v2/2) * gamma((v1+v2)/2))

  for `df1, df2, nc > 0`.

  `ncf` takes `df1`, `df2` and `nc` as shape parameters.

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `ncf.pdf(x, dfn, dfd, nc, loc, scale)` is identically equivalent to `ncf.pdf(y, dfn, dfd, nc) / scale` with `y = (x - loc) / scale`.

  Examples

      >>> from scipy.stats import ncf
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Calculate a few first moments:

      >>> dfn, dfd, nc = 27, 27, 0.416
      >>> mean, var, skew, kurt = ncf.stats(dfn, dfd, nc, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(ncf.ppf(0.01, dfn, dfd, nc),
      ...                 ncf.ppf(0.99, dfn, dfd, nc), 100)
      >>> ax.plot(x, ncf.pdf(x, dfn, dfd, nc),
      ...        'r-', lw=5, alpha=0.6, label='ncf 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 = ncf(dfn, dfd, nc)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = ncf.ppf([0.001, 0.5, 0.999], dfn, dfd, nc)
      >>> np.allclose([0.001, 0.5, 0.999], ncf.cdf(vals, dfn, dfd, nc))
      True

  Generate random numbers:

      >>> r = ncf.rvs(dfn, dfd, nc, size=1000)

  And compare the histogram:

      >>> ax.hist(r, normed=True, histtype='stepfilled', alpha=0.2)
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()

  (Source code)

  Methods

  `rvs(dfn, dfd, nc, loc=0, scale=1, size=1, random_state=None)` | Random variates.
  `pdf(x, dfn, dfd, nc, loc=0, scale=1)` | Probability density function.
  `logpdf(x, dfn, dfd, nc, loc=0, scale=1)` | Log of the probability density function.
  `cdf(x, dfn, dfd, nc, loc=0, scale=1)` | Cumulative density function.
  `logcdf(x, dfn, dfd, nc, loc=0, scale=1)` | Log of the cumulative density function.
  `sf(x, dfn, dfd, nc, loc=0, scale=1)` | Survival function (`1 - cdf` — sometimes more accurate).
  `logsf(x, dfn, dfd, nc, loc=0, scale=1)` | Log of the survival function.
  `ppf(q, dfn, dfd, nc, loc=0, scale=1)` | Percent point function (inverse of `cdf` — percentiles).
  `isf(q, dfn, dfd, nc, loc=0, scale=1)` | Inverse survival function (inverse of `sf`).
  `moment(n, dfn, dfd, nc, loc=0, scale=1)` | Non-central moment of order n
  `stats(dfn, dfd, nc, loc=0, scale=1, moments='mv')` | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  `entropy(dfn, dfd, nc, loc=0, scale=1)` | (Differential) entropy of the RV.
  `fit(data, dfn, dfd, nc, loc=0, scale=1)` | Parameter estimates for generic data.
  `expect(func, dfn, dfd, nc, 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, nc, loc=0, scale=1)` | Median of the distribution.
  `mean(dfn, dfd, nc, loc=0, scale=1)` | Mean of the distribution.
  `var(dfn, dfd, nc, loc=0, scale=1)` | Variance of the distribution.
  `std(dfn, dfd, nc, loc=0, scale=1)` | Standard deviation of the distribution.
  `interval(alpha, dfn, dfd, nc, loc=0, scale=1)` | Endpoints of the range that contains alpha percent of the distribution