Statistics.cpp
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	#include <Statistics.hpp>
	#include <cmath> // M_PI, pow(), sqrt(), log(), lgamma()
	#include <boost/math/distributions.hpp> // beta_distribution
	#include <boost/random.hpp>


	double normal_pmf(double x, double mean, double sd)
	{ static double pi_2 = 2.*M_PI ;
	return ( 1. / ( sd * sqrt(pi_2) )) * exp(-0.5 * pow((x-mean)/sd, 2.0 ) );
	}

	double poisson_pmf(int x, double lambda)
	{ if((x != 0) and (lambda == 0))
	{ return 0. ; }
	else if((x == 0) and (lambda == 0))
	{ return 1. ; }

	// if(x + lambda == 0)
	// { return 1. ; }
	return exp(x * log(lambda) - lgamma(x + 1.0) - lambda);
	}

	double poisson_cdf(int x, double lambda, bool lower_tail)
	{ double prob = 1. ;
	if(lambda != 0)
	{ boost::math::poisson_distribution<> pois_dist(lambda) ;
	prob = cdf(pois_dist, x) ;
	}
	// prob of an observation as big as x
	if(lower_tail)
	{ return prob ; }
	// prob of an observation at least as big as x (pvalue)
	else
	{ return 1 - prob ; }
	}

	double beta_pmf(double x, double alpha, double beta)
	{
	boost::math::beta_distribution<> beta_dist(alpha, beta) ;
	double y = quantile(beta_dist, x) ;

	return y ;
	}