A non-central F distribution continuous random variable.

Continuous random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:


x : array-like


q : array-like

lower or upper tail probability

dfn,dfd,nc : array-like

shape parameters

loc : array-like, optional

location parameter (default=0)

scale : array-like, optional

scale parameter (default=1)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : string, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)


ncf.rvs(dfn,dfd,nc,loc=0,scale=1,size=1) :

  • random variates

ncf.pdf(x,dfn,dfd,nc,loc=0,scale=1) :

  • probability density function

ncf.cdf(x,dfn,dfd,nc,loc=0,scale=1) :

  • cumulative density function

ncf.sf(x,dfn,dfd,nc,loc=0,scale=1) :

  • survival function (1-cdf — sometimes more accurate)

ncf.ppf(q,dfn,dfd,nc,loc=0,scale=1) :

  • percent point function (inverse of cdf — percentiles)

ncf.isf(q,dfn,dfd,nc,loc=0,scale=1) :

  • inverse survival function (inverse of sf)

ncf.stats(dfn,dfd,nc,loc=0,scale=1,moments=’mv’) :

  • mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’)

ncf.entropy(dfn,dfd,nc,loc=0,scale=1) :

  • (differential) entropy of the RV.,dfn,dfd,nc,loc=0,scale=1) :

  • Parameter estimates for ncf data

Alternatively, the object may be called (as a function) to fix the shape, :

location, and scale parameters returning a “frozen” continuous RV object: :

rv = ncf(dfn,dfd,nc,loc=0,scale=1) :

  • frozen RV object with the same methods but holding the given shape, location, and scale fixed


>>> import matplotlib.pyplot as plt
>>> numargs = ncf.numargs
>>> [ dfn,dfd,nc ] = [0.9,]*numargs
>>> rv = ncf(dfn,dfd,nc)

Display frozen pdf

>>> x = np.linspace(0,np.minimum(rv.dist.b,3))
>>> h=plt.plot(x,rv.pdf(x))

Check accuracy of cdf and ppf

>>> prb = ncf.cdf(x,dfn,dfd,nc)
>>> h=plt.semilogy(np.abs(x-ncf.ppf(prb,c))+1e-20)

Random number generation

>>> R = ncf.rvs(dfn,dfd,nc,size=100)

Non-central F distribution

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.

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