ó î&]\c@`sŽddlmZmZmZddlmZddlmZmZm Z m Z ddl m Z ddlmZddlmZmZmZmZmZmZmZmZmZmZddlZdd lmZmZmZm Z d efd „ƒYZ!e!d d ƒZ"de!fd„ƒYZ#e#ddd dƒZ$defd„ƒYZ%e%d dƒZ&defd„ƒYZ'e'ddd dddƒZ(defd„ƒYZ)e)d dƒZ*defd„ƒYZ+e+ddd d dd!ƒZ,d"efd#„ƒYZ-e-d d$dd%ƒZ.d&efd'„ƒYZ/e/ddd d(dd)ƒZ0d*efd+„ƒYZ1e1d d,dd-ƒZ2d.efd/„ƒYZ3e3d d0dd1ƒZ4d2efd3„ƒYZ5e5ddd d4dd5ƒZ6d6efd7„ƒYZ7e7dej8 d d8dd9ƒZ9d:efd;„ƒYZ:e:dej8 d d<dd=ƒZ;d>efd?„ƒYZ<e<d d@ddƒZ=e>e?ƒj@ƒƒZAe eAeƒ\ZBZCeBeCZDdS(Ai(tdivisiontprint_functiontabsolute_import(tspecial(tentrt logsumexptbetalntgammaln(t broadcast_to(t _lazywhere( tfloortceiltlogtexptsqrttlog1ptexpm1ttanhtcoshtsinhNi(t rv_discretet _ncx2_pdft _ncx2_cdftget_distribution_namest binom_gencB`sbeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z dd „Z d „Z RS( sEA 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 cC`s|jj|||jƒS(N(t _random_statetbinomialt_size(tselftntp((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_rvs*scC`s'||_|dk|dk@|dk@S(Nii(tb(RRR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt _argcheck-s cC`set|ƒ}t|dƒt|dƒt||dƒ}|tj||ƒtj||| ƒS(Ni(R tgamlnRtxlogytxlog1py(RtxRRtktcombiln((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_logpmf1s 0cC`st|j|||ƒƒS(N(R R((RR%RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_pmf6scC`s%t|ƒ}tj|||ƒ}|S(N(R Rtbdtr(RR%RRR&tvals((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_cdf:s cC`st|ƒ}tj|||ƒS(N(R Rtbdtrc(RR%RRR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_sf?s cC`s_ttj|||ƒƒ}tj|ddƒ}tj|||ƒ}tj||k||ƒS(Nii(R RtbdtriktnptmaximumR*twhere(RtqRRR+tvals1ttemp((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_ppfCstmvc C`s†d|}||}|||}d\}}d|krQ||t|ƒ}nd|krvdd|||}n||||fS(Ngð?tsR&i(NN(tNoneR( RRRtmomentsR3tmutvartg1tg2((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_statsIs     cC`sBtjd|d!}|j|||ƒ}tjt|ƒddƒS(Niitaxis(R0tr_R)tsumR(RRRR&R+((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_entropyTs( t__name__t __module__t__doc__RR!R(R)R,R.R6R?RC(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRs        tnametbinomt bernoulli_gencB`s_eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z RS( sA Bernoulli discrete random variable. %(before_notes)s Notes ----- The probability mass function for `bernoulli` is: .. math:: f(k) = \begin{cases}1-p &\text{if } k = 0\\ p &\text{if } k = 1\end{cases} for :math:`k` in :math:`\{0, 1\}`. `bernoulli` takes :math:`p` as shape parameter. %(after_notes)s %(example)s cC`stj|d|ƒS(Ni(RR(RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRtscC`s|dk|dk@S(Nii((RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!wscC`stj|d|ƒS(Ni(RHR((RR%R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(zscC`stj|d|ƒS(Ni(RHR)(RR%R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)}scC`stj|d|ƒS(Ni(RHR,(RR%R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,‚scC`stj|d|ƒS(Ni(RHR.(RR%R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.…scC`stj|d|ƒS(Ni(RHR6(RR3R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6ˆscC`stjd|ƒS(Ni(RHR?(RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?‹scC`st|ƒtd|ƒS(Ni(R(RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRCŽs( RDRERFRR!R(R)R,R.R6R?RC(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRI]s        R t bernoullit nbinom_gencB`sVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( s_A negative binomial discrete random variable. %(before_notes)s Notes ----- Negative binomial distribution describes a sequence of i.i.d. Bernoulli trials, repeated until a predefined, non-random number of successes occurs. The probability mass function of the number of failures for `nbinom` is: .. math:: f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k for :math:`k \ge 0`. `nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the number of successes, whereas p is the probability of a single success. %(after_notes)s %(example)s cC`s|jj|||jƒS(N(Rtnegative_binomialR(RRR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR¯scC`s|dk|dk@|dk@S(Nii((RRR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!²scC`st|j|||ƒƒS(N(R R((RR%RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)µscC`sKt||ƒt|dƒt|ƒ}||t|ƒtj|| ƒS(Ni(R"R RR$(RR%RRtcoeff((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(¹s(cC`s#t|ƒ}tj||d|ƒS(Ni(R Rtbetainc(RR%RRR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,½s cC`st|ƒ}tj|||ƒS(N(R Rtnbdtrc(RR%RRR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_sf_skipÁs cC`sbttj|||ƒƒ}|djdtjƒ}|j|||ƒ}tj||k||ƒS(Nig(R RtnbdtriktclipR0tinfR,R2(RR3RRR+R4R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6Æsc C`svd|}|d}||}|||}||t|||ƒ}dd|||||}||||fS(Ngð?i(R( RRRtQtPR;R<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?Ìs   ( RDRERFRR!R)R(R,RPR6R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRK•s       tnbinomtgeom_gencB`s_eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z RS( s$A geometric discrete random variable. %(before_notes)s Notes ----- The probability mass function for `geom` is: .. math:: f(k) = (1-p)^{k-1} p for :math:`k \ge 1`. `geom` takes :math:`p` as shape parameter. %(after_notes)s %(example)s cC`s|jj|d|jƒS(Ntsize(Rt geometricR(RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRïscC`s|dk|dk@S(Nii((RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!òscC`stjd||dƒ|S(Ni(R0tpower(RR&R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)õscC`stj|d| ƒt|ƒS(Ni(RR$R (RR&R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(ùscC`s"t|ƒ}tt| ƒ|ƒ S(N(R RR(RR%RR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,üs cC`stj|j||ƒƒS(N(R0R t_logsf(RR%R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.scC`st|ƒ}|t| ƒS(N(R R(RR%RR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR[s cC`s[tt| ƒt| ƒƒ}|j|d|ƒ}tj||k|dk@|d|ƒS(Nii(R RR,R0R2(RR3RR+R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6scC`sid|}d|}|||}d|t|ƒ}tjdddg|ƒd|}||||fS(Ngð?g@iiúÿÿÿi(RR0tpolyval(RRR;tqrR<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR? s   #( RDRERFRR!R)R(R,R.R[R6R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRWÙs        tatgeomtlongnames A geometrict hypergeom_gencB`sVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( s³A hypergeometric discrete random variable. The hypergeometric distribution models drawing objects from a bin. `M` is the total number of objects, `n` is total number of Type I objects. The random variate represents the number of Type I objects in `N` drawn without replacement from the total population. %(before_notes)s Notes ----- The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not universally accepted. See the Examples for a clarification of the definitions used here. The probability mass function is defined as, .. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}} {\binom{M}{N}} for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial coefficients are defined as, .. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}. %(after_notes)s Examples -------- >>> from scipy.stats import hypergeom >>> import matplotlib.pyplot as plt Suppose we have a collection of 20 animals, of which 7 are dogs. Then if we want to know the probability of finding a given number of dogs if we choose at random 12 of the 20 animals, we can initialize a frozen distribution and plot the probability mass function: >>> [M, n, N] = [20, 7, 12] >>> rv = hypergeom(M, n, N) >>> x = np.arange(0, n+1) >>> pmf_dogs = rv.pmf(x) >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.plot(x, pmf_dogs, 'bo') >>> ax.vlines(x, 0, pmf_dogs, lw=2) >>> ax.set_xlabel('# of dogs in our group of chosen animals') >>> ax.set_ylabel('hypergeom PMF') >>> plt.show() Instead of using a frozen distribution we can also use `hypergeom` methods directly. To for example obtain the cumulative distribution function, use: >>> prb = hypergeom.cdf(x, M, n, N) And to generate random numbers: >>> R = hypergeom.rvs(M, n, N, size=10) cC`s#|jj||||d|jƒS(NRX(RthypergeometricR(RtMRtN((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRVscC`sp|dk|dk@|dk@}|||k||k@M}tj|||dƒ|_tj||ƒ|_|S(Ni(R0R1R^tminimumR (RRcRRdtcond((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!Ys  c C`s£||}}||}t|ddƒt|ddƒt||d|dƒt|d||dƒt||d|||dƒt|ddƒ}|S(Ni(R( RR&RcRRdttottgoodtbadtresult((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(`s   tcC`st|j||||ƒƒS(N(R R((RR&RcRRd((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)hsc C`sLd|d|d|}}}||}||}||}|||||d|||d}|||d||dt|d|||||ƒ}||dd|||d||} | |d||9} | d|||||d|d7} | ||||||d|d} |||| fS( Ngð?iig@g@g@ig@(R( RRcRRdtmRR;R<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?ms    *@***cC`sVtj|||t||ƒd!}|j||||ƒ}tjt|ƒddƒS(NiR@i(R0RAtmintpmfRBR(RRcRRdR&R+((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRC~s%c C`sg}xkt||||ƒD]T\}}}} tj|d| dƒ} |jtj|j| ||| ƒƒƒqWtj|ƒS(s1More precise calculation, 1 - cdf doesn't cut it.i(tzipR0tarangetappendRBR)tasarray( RR&RcRRdtrestquantRgRhtdrawtk2((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.ƒs (,c C`s~g}xht||||ƒD]Q\}}}} tj|d| dƒ} |jt|j| ||| ƒƒƒqWtj|ƒS(s7 More precise calculation than log(sf) i(RnR0RoRpRR(Rq( RR&RcRRdRrRsRgRhRtRu((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR[‘s ()( RDRERFRR!R(R)R?RCR.R[(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRas=       t hypergeomt logser_gencB`s2eZdZd„Zd„Zd„Zd„ZRS(sKA Logarithmic (Log-Series, Series) discrete random variable. %(before_notes)s Notes ----- The probability mass function for `logser` is: .. math:: f(k) = - \frac{p^k}{k \log(1-p)} for :math:`k \ge 1`. `logser` takes :math:`p` as shape parameter. %(after_notes)s %(example)s cC`s|jj|d|jƒS(NRX(Rt logseriesR(RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR·scC`s|dk|dk@S(Nii((RR((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!¼scC`s'tj||ƒ d|tj| ƒS(Ngð?(R0RZRR(RR&R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)¿sc C`s1tj| ƒ}||d|}| ||dd}|||}| |d|d|d}|d||d|d}|tj|dƒ}| |d|ddd||ddd|||dd} | d||d|||d|d} | |dd} |||| fS( Ngð?iigø?iiig@(RRR0RZ( RRtrR;tmu2pR<tmu3ptmu3R=tmu4ptmu4R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?Ãs?.(RDRERFRR!R)R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRw¡s    tlogsers A logarithmict poisson_gencB`sVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( s8A Poisson discrete random variable. %(before_notes)s Notes ----- The probability mass function for `poisson` is: .. math:: f(k) = \exp(-\mu) \frac{\mu^k}{k!} for :math:`k \ge 0`. `poisson` takes :math:`\mu` as shape parameter. %(after_notes)s %(example)s cC`s |dkS(Ni((RR;((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!îscC`s|jj||jƒS(N(RtpoissonR(RR;((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRñscC`s(tj||ƒt|dƒ|}|S(Ni(RR#R"(RR&R;tPk((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(ôs$cC`st|j||ƒƒS(N(R R((RR&R;((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)øscC`st|ƒ}tj||ƒS(N(R Rtpdtr(RR%R;R&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,üs cC`st|ƒ}tj||ƒS(N(R Rtpdtrc(RR%R;R&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.s cC`sYttj||ƒƒ}tj|ddƒ}tj||ƒ}tj||k||ƒS(Nii(R RtpdtrikR0R1RƒR2(RR3R;R+R4R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6scC`sm|}tj|ƒ}|dk}t||fd„tjƒ}t||fd„tjƒ}||||fS(NicS`std|ƒS(Ngð?(R(R%((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pytscS`sd|S(Ngð?((R%((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR†s(R0RqR RS(RR;R<ttmpt mu_nonzeroR=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR? s  ( RDRERFR!RR(R)R,R.R6R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR€Ös       Rs A Poissont planck_gencB`sVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( seA Planck discrete exponential random variable. %(before_notes)s Notes ----- The probability mass function for `planck` is: .. math:: f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) for :math:`k \ge 0` and :math:`\lambda > 0`. `planck` takes :math:`\lambda` as shape parameter. %(after_notes)s %(example)s cC`s |dkS(Ni((Rtlambda_((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!,scC`sdt| ƒt| |ƒS(Ni(R (RR&RŠ((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)/scC`s#t|ƒ}dt| |dƒS(Ni(R R (RR%RŠR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,2s cC`stj|j||ƒƒS(N(R0R R[(RR%RŠ((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.6scC`st|ƒ}| |dS(Ni(R (RR%RŠR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR[9s cC`sftd|t| ƒdƒ}|dj|jtjƒ}|j||ƒ}tj||k||ƒS(Ngð¿i(R RRRR^R0RSR,R2(RR3RŠR+R4R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6=scC`shdt|ƒd}t| ƒt| ƒd}dt|dƒ}ddt|ƒ}||||fS(Niig@i(R RR(RRŠR;R<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?Cs cC`s4|}dt| ƒ}|t| ƒ|t|ƒS(Ni(R R (RRŠtltC((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRCJs( RDRERFR!R)R,R.R[R6R?RC(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR‰s       tplancksA discrete exponential t boltzmann_gencB`s;eZdZd„Zd„Zd„Zd„Zd„ZRS(s—A Boltzmann (Truncated Discrete Exponential) random variable. %(before_notes)s Notes ----- The probability mass function for `boltzmann` is: .. math:: f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) / (1-\exp(-\lambda N)) for :math:`k = 0,..., N-1`. `boltzmann` takes :math:`\lambda > 0` and :math:`N > 0` as shape parameters. %(after_notes)s %(example)s cC`s*d|_|d|_|dk|dk@S(Nii(R^R (RRŠRd((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!is  cC`s7dt| ƒdt| |ƒ}|t| |ƒS(Ni(R (RR&RŠRdtfact((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)ns$cC`s6t|ƒ}dt| |dƒdt| |ƒS(Ni(R R (RR%RŠRdR&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,ts cC`s‚|dt| |ƒ}td|td|ƒdƒ}|djdtjƒ}|j|||ƒ}tj||k||ƒS(Nigð¿g(R R R RRR0RSR,R2(RR3RŠRdtqnewR+R4R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6xs "c C`s*t| ƒ}t| |ƒ}|d|||d|}|d|d|||d|d}d|d|}||d|||}|d||d|d|d|} | |d} |dd||||d|d|dd|||} | ||} ||| | fS(Ngð?iiigø?i(R ( RRŠRdtztzNR;R<ttrmttrm2R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?s **B(RDRERFR!R)R,R6R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRŽSs     t boltzmanns!A truncated discrete exponential t randint_gencB`sMeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z RS(sFA uniform discrete random variable. %(before_notes)s Notes ----- The probability mass function for `randint` is: .. math:: f(k) = \frac{1}{high - low} for ``k = low, ..., high - 1``. `randint` takes ``low`` and ``high`` as shape parameters. %(after_notes)s %(example)s cC`s ||_|d|_||kS(Ni(R^R (Rtlowthigh((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!§s  cC`s:tj|ƒ||}tj||k||k@|dƒS(Ng(R0t ones_likeR2(RR&R—R˜R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)¬scC`s t|ƒ}||d||S(Ngð?(R (RR%R—R˜R&((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,±s cC`s`t||||ƒd}|dj||ƒ}|j|||ƒ}tj||k||ƒS(Ni(R RRR,R0R2(RR3R—R˜R+R4R5((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6µsc C`stj|ƒtj|ƒ}}||dd}||}||dd}d}d||d||d} |||| fS( Ngð?iig(@ggÀg@g333333ó¿(R0Rq( RR—R˜tm2tm1R;tdR<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?»s cC`sd|jdk r6t||jƒ}t||jƒ}ntj|jjdtjgƒ}|||ƒS(s=An array of *size* random integers >= ``low`` and < ``high``.totypesN(RR9RR0t vectorizeRtrandinttint_(RR—R˜RŸ((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRÄs !cC`st||ƒS(N(R (RR—R˜((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRCÐs( RDRERFR!R)R,R6R?RRC(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR–‘s     RŸs#A discrete uniform (random integer)tzipf_gencB`s2eZdZd„Zd„Zd„Zd„ZRS(srA Zipf discrete random variable. %(before_notes)s Notes ----- The probability mass function for `zipf` is: .. math:: f(k, a) = \frac{1}{\zeta(a) k^a} for :math:`k \ge 1`. `zipf` takes :math:`a` as shape parameter. :math:`\zeta` is the Riemann zeta function (`scipy.special.zeta`) %(after_notes)s %(example)s cC`s|jj|d|jƒS(NRX(RtzipfR(RR^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRðscC`s |dkS(Ni((RR^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!óscC`s"dtj|dƒ||}|S(Ngð?i(Rtzeta(RR&R^R‚((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)öscC`s)t||dk||fd„tjƒS(NicS`s$tj||dƒtj|dƒS(Ni(RR£(R^R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR†þs(R R0RS(RRR^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyt_munpûs(RDRERFRR!R)R¤(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR¡Ùs    R¢sA Zipft dlaplace_gencB`s;eZdZd„Zd„Zd„Zd„Zd„ZRS(sLA Laplacian discrete random variable. %(before_notes)s Notes ----- The probability mass function for `dlaplace` is: .. math:: f(k) = \tanh(a/2) \exp(-a |k|) for integers :math:`k` and :math:`a > 0`. `dlaplace` takes :math:`a` as shape parameter. %(after_notes)s %(example)s cC`s#t|dƒt| t|ƒƒS(Ng@(RR tabs(RR&R^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)scC`sCt|ƒ}d„}d„}t|dk||fd|d|ƒS(NcS`s!dt| |ƒt|ƒdS(Ngð?i(R (R&R^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR†!scS`s t||dƒt|ƒdS(Ni(R (R&R^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR†"sitftf2(R R (RR%R^R&R§R¨((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,s   cC`s”dt|ƒ}ttj|ddt| ƒkt||ƒ|dtd||ƒ |ƒƒ}|d}tj|j||ƒ|k||ƒS(Nigð?(R R R0R2R R,(RR3R^tconstR+R4((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR6%s ! cC`sht|ƒ}d||dd}d||dd|d|dd}d|d||ddfS(Ng@gð?ig$@igg@(R (RR^teatmu2R~((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?-s *cC`s"|t|ƒtt|dƒƒS(Ng@(RR R(RR^((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRC3s(RDRERFR)R,R6R?RC(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR¥s     tdlaplacesA discrete Laplaciant skellam_gencB`s2eZdZd„Zd„Zd„Zd„ZRS(sÍA Skellam discrete random variable. %(before_notes)s Notes ----- Probability distribution of the difference of two correlated or uncorrelated Poisson random variables. Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with expected values :math:`\lambda_1` and :math:`\lambda_2`. Then, :math:`k_1 - k_2` follows a Skellam distribution with parameters :math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and :math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where :math:`\rho` is the correlation coefficient between :math:`k_1` and :math:`k_2`. If the two Poisson-distributed r.v. are independent then :math:`\rho = 0`. Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive. For details see: https://en.wikipedia.org/wiki/Skellam_distribution `skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters. %(after_notes)s %(example)s cC`s/|j}|jj||ƒ|jj||ƒS(N(RRR(Rtmu1R«R((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyRYs cC`s_tj|dktd|dd|d|ƒdtd|dd|d|ƒdƒ}|S(Niii(R0R2R(RR%R®R«tpx((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)^s#)c C`sct|ƒ}tj|dktd|d|d|ƒdtd|d|dd|ƒƒ}|S(Niiiþÿÿÿi(R R0R2R(RR%R®R«R¯((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,es  )cC`sB||}||}|t|dƒ}d|}||||fS(Nii(R(RR®R«tmeanR<R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?ls    (RDRERFRR)R,R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR­;s    tskellams A Skellamt yulesimon_gencB`sVeZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z RS( sîA Yule-Simon discrete random variable. %(before_notes)s Notes ----- The probability mass function for the `yulesimon` is: .. math:: f(k) = \alpha B(k, \alpha+1) for :math:`k=1,2,3,...`, where :math:`\alpha>0`. Here :math:`B` refers to the `scipy.special.beta` function. The sampling of random variates is based on pg 553, Section 6.3 of [1]_. Our notation maps to the referenced logic via :math:`\alpha=a-1`. For details see the wikipedia entry [2]_. References ---------- .. [1] Devroye, Luc. "Non-uniform Random Variate Generation", (1986) Springer, New York. .. [2] https://en.wikipedia.org/wiki/Yule-Simon_distribution %(after_notes)s %(example)s cC`sQ|jj|jƒ}|jj|jƒ}t| tt| |ƒ ƒƒ}|S(N(Rtstandard_exponentialRR RR (RtalphatE1tE2tans((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR™s#cC`s|tj||dƒS(Ni(Rtbeta(RR%R´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR)ŸscC`s |dkS(Ni((RR´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR!¢scC`st|ƒtj||dƒS(Ni(R RR(RR%R´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR(¥scC`sd|tj||dƒS(Ni(RR¸(RR%R´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR,¨scC`s|tj||dƒS(Ni(RR¸(RR%R´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR.«scC`st|ƒtj||dƒS(Ni(R RR(RR%R´((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR[®scC`sPtj|dktj||dƒ}tj|dk|d|d|ddtjƒ}tj|dktj|ƒ}tj|dkt|dƒ|dd||dtjƒ}tj|dktj|ƒ}tj|dk|d|dd|d||d|dtjƒ}tj|dktj|ƒ}||||fS(Niig@iii1i(R0R2RStnanR(RR´R;R«R=R>((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR?±s& % !( RDRERFRR)R!R(R,R.R[R?(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pyR²ws!       t yulesimon(Et __future__RRRtscipyRt scipy.specialRRRRR"tscipy._lib._numpy_compatRtscipy._lib._utilR tnumpyR R R R RRRRRRR0t_distn_infrastructureRRRRRRHRIRJRKRVRWR_RaRvRwRR€RR‰RRŽR•R–RŸR¡R¢R¥RSR¬R­R±R²Rºtlisttglobalstitemstpairst _distn_namest_distn_gen_namest__all__(((s;lib/python2.7/site-packages/scipy/stats/_discrete_distns.pytsP"F "F5A<…2=::  C)29K