[section:binomial_dist Binomial Distribution] ``#include `` namespace boost{ namespace math{ template class binomial_distribution; typedef binomial_distribution<> binomial; template class binomial_distribution { public: typedef RealType value_type; typedef Policy policy_type; static const ``['unspecified-type]`` clopper_pearson_exact_interval; static const ``['unspecified-type]`` jeffreys_prior_interval; // construct: binomial_distribution(RealType n, RealType p); // parameter access:: RealType success_fraction() const; RealType trials() const; // Bounds on success fraction: static RealType find_lower_bound_on_p( RealType trials, RealType successes, RealType probability, ``['unspecified-type]`` method = clopper_pearson_exact_interval); static RealType find_upper_bound_on_p( RealType trials, RealType successes, RealType probability, ``['unspecified-type]`` method = clopper_pearson_exact_interval); // estimate min/max number of trials: static RealType find_minimum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha); // risk level static RealType find_maximum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha); // risk level }; }} // namespaces The class type `binomial_distribution` represents a [@http://mathworld.wolfram.com/BinomialDistribution.html binomial distribution]: it is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are labelled "success" and "failure". The __binomial_distrib is used to obtain the probability of observing k successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials. [note The random variable for the binomial distribution is the number of successes, (the number of trials is a fixed property of the distribution) whereas for the negative binomial, the random variable is the number of trials, for a fixed number of successes.] The PDF for the binomial distribution is given by: [equation binomial_ref2] The following two graphs illustrate how the PDF changes depending upon the distributions parameters, first we'll keep the success fraction /p/ fixed at 0.5, and vary the sample size: [graph binomial_pdf_1] Alternatively, we can keep the sample size fixed at N=20 and vary the success fraction /p/: [graph binomial_pdf_2] [discrete_quantile_warning Binomial] [h4 Member Functions] [h5 Construct] binomial_distribution(RealType n, RealType p); Constructor: /n/ is the total number of trials, /p/ is the probability of success of a single trial. Requires `0 <= p <= 1`, and `n >= 0`, otherwise calls __domain_error. [h5 Accessors] RealType success_fraction() const; Returns the parameter /p/ from which this distribution was constructed. RealType trials() const; Returns the parameter /n/ from which this distribution was constructed. [h5 Lower Bound on the Success Fraction] static RealType find_lower_bound_on_p( RealType trials, RealType successes, RealType alpha, ``['unspecified-type]`` method = clopper_pearson_exact_interval); Returns a lower bound on the success fraction: [variablelist [[trials][The total number of trials conducted.]] [[successes][The number of successes that occurred.]] [[alpha][The largest acceptable probability that the true value of the success fraction is [*less than] the value returned.]] [[method][An optional parameter that specifies the method to be used to compute the interval (See below).]] ] For example, if you observe /k/ successes from /n/ trials the best estimate for the success fraction is simply ['k/n], but if you want to be 95% sure that the true value is [*greater than] some value, ['p[sub min]], then: p``[sub min]`` = binomial_distribution::find_lower_bound_on_p(n, k, 0.05); [link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.] There are currently two possible values available for the /method/ optional parameter: /clopper_pearson_exact_interval/ or /jeffreys_prior_interval/. These constants are both members of class template `binomial_distribution`, so usage is for example: p = binomial_distribution::find_lower_bound_on_p( n, k, 0.05, binomial_distribution::jeffreys_prior_interval); The default method if this parameter is not specified is the Clopper Pearson "exact" interval. This produces an interval that guarantees at least `100(1-alpha)%` coverage, but which is known to be overly conservative, sometimes producing intervals with much greater than the requested coverage. The alternative calculation method produces a non-informative Jeffreys Prior interval. It produces `100(1-alpha)%` coverage only ['in the average case], though is typically very close to the requested coverage level. It is one of the main methods of calculation recommended in the review by Brown, Cai and DasGupta. Please note that the "textbook" calculation method using a normal approximation (the Wald interval) is deliberately not provided: it is known to produce consistently poor results, even when the sample size is surprisingly large. Refer to Brown, Cai and DasGupta for a full explanation. Many other methods of calculation are available, and may be more appropriate for specific situations. Unfortunately there appears to be no consensus amongst statisticians as to which is "best": refer to the discussion at the end of Brown, Cai and DasGupta for examples. The two methods provided here were chosen principally because they can be used for both one and two sided intervals. See also: Lawrence D. Brown, T. Tony Cai and Anirban DasGupta (2001), Interval Estimation for a Binomial Proportion, Statistical Science, Vol. 16, No. 2, 101-133. T. Tony Cai (2005), One-sided confidence intervals in discrete distributions, Journal of Statistical Planning and Inference 131, 63-88. Agresti, A. and Coull, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. Amer. Statist. 52 119-126. Clopper, C. J. and Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26 404-413. [h5 Upper Bound on the Success Fraction] static RealType find_upper_bound_on_p( RealType trials, RealType successes, RealType alpha, ``['unspecified-type]`` method = clopper_pearson_exact_interval); Returns an upper bound on the success fraction: [variablelist [[trials][The total number of trials conducted.]] [[successes][The number of successes that occurred.]] [[alpha][The largest acceptable probability that the true value of the success fraction is [*greater than] the value returned.]] [[method][An optional parameter that specifies the method to be used to compute the interval. Refer to the documentation for `find_upper_bound_on_p` above for the meaning of the method options.]] ] For example, if you observe /k/ successes from /n/ trials the best estimate for the success fraction is simply ['k/n], but if you want to be 95% sure that the true value is [*less than] some value, ['p[sub max]], then: p``[sub max]`` = binomial_distribution::find_upper_bound_on_p(n, k, 0.05); [link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.] [note In order to obtain a two sided bound on the success fraction, you call both `find_lower_bound_on_p` *and* `find_upper_bound_on_p` each with the same arguments. If the desired risk level that the true success fraction lies outside the bounds is [alpha], then you pass [alpha]/2 to these functions. So for example a two sided 95% confidence interval would be obtained by passing [alpha] = 0.025 to each of the functions. [link math_toolkit.stat_tut.weg.binom_eg.binom_conf See worked example.] ] [h5 Estimating the Number of Trials Required for a Certain Number of Successes] static RealType find_minimum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha); // probability threshold This function estimates the minimum number of trials required to ensure that more than k events is observed with a level of risk /alpha/ that k or fewer events occur. [variablelist [[k][The number of success observed.]] [[p][The probability of success for each trial.]] [[alpha][The maximum acceptable probability that k events or fewer will be observed.]] ] For example: binomial_distribution::find_number_of_trials(10, 0.5, 0.05); Returns the smallest number of trials we must conduct to be 95% sure of seeing 10 events that occur with frequency one half. [h5 Estimating the Maximum Number of Trials to Ensure no more than a Certain Number of Successes] static RealType find_maximum_number_of_trials( RealType k, // number of events RealType p, // success fraction RealType alpha); // probability threshold This function estimates the maximum number of trials we can conduct to ensure that k successes or fewer are observed, with a risk /alpha/ that more than k occur. [variablelist [[k][The number of success observed.]] [[p][The probability of success for each trial.]] [[alpha][The maximum acceptable probability that more than k events will be observed.]] ] For example: binomial_distribution::find_maximum_number_of_trials(0, 1e-6, 0.05); Returns the largest number of trials we can conduct and still be 95% certain of not observing any events that occur with one in a million frequency. This is typically used in failure analysis. [link math_toolkit.stat_tut.weg.binom_eg.binom_size_eg See Worked Example.] [h4 Non-member Accessors] All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions] that are generic to all distributions are supported: __usual_accessors. The domain for the random variable /k/ is `0 <= k <= N`, otherwise a __domain_error is returned. It's worth taking a moment to define what these accessors actually mean in the context of this distribution: [table Meaning of the non-member accessors [[Function][Meaning]] [[__pdf] [The probability of obtaining [*exactly k successes] from n trials with success fraction p. For example: `pdf(binomial(n, p), k)`]] [[__cdf] [The probability of obtaining [*k successes or fewer] from n trials with success fraction p. For example: `cdf(binomial(n, p), k)`]] [[__ccdf] [The probability of obtaining [*more than k successes] from n trials with success fraction p. For example: `cdf(complement(binomial(n, p), k))`]] [[__quantile] [Given a binomial distribution with ['n] trials, success fraction ['p] and probability ['P], finds the largest number of successes ['k] whose CDF is less than ['P]. It is strongly recommended that you read the tutorial [link math_toolkit.pol_tutorial.understand_dis_quant Understanding Quantiles of Discrete Distributions] before using the quantile function.]] [[__quantile_c] [Given a binomial distribution with ['n] trials, success fraction ['p] and probability ['Q], finds the smallest number of successes ['k] whose CDF is greater than ['1-Q]. It is strongly recommended that you read the tutorial [link math_toolkit.pol_tutorial.understand_dis_quant Understanding Quantiles of Discrete Distributions] before using the quantile function.]] ] [h4 Examples] Various [link math_toolkit.stat_tut.weg.binom_eg worked examples] are available illustrating the use of the binomial distribution. [h4 Accuracy] This distribution is implemented using the incomplete beta functions __ibeta and __ibetac, please refer to these functions for information on accuracy. [h4 Implementation] In the following table /p/ is the probability that one trial will be successful (the success fraction), /n/ is the number of trials, /k/ is the number of successes, /p/ is the probability and /q = 1-p/. [table [[Function][Implementation Notes]] [[pdf][Implementation is in terms of __ibeta_derivative: if [sub n]C[sub k ] is the binomial coefficient of a and b, then we have: [equation binomial_ref1] Which can be evaluated as `ibeta_derivative(k+1, n-k+1, p) / (n+1)` The function __ibeta_derivative is used here, since it has already been optimised for the lowest possible error - indeed this is really just a thin wrapper around part of the internals of the incomplete beta function. There are also various special cases: refer to the code for details. ]] [[cdf][Using the relation: `` p = I[sub 1-p](n - k, k + 1) = 1 - I[sub p](k + 1, n - k) = __ibetac(k + 1, n - k, p)`` There are also various special cases: refer to the code for details. ]] [[cdf complement][Using the relation: q = __ibeta(k + 1, n - k, p) There are also various special cases: refer to the code for details. ]] [[quantile][Since the cdf is non-linear in variate /k/ none of the inverse incomplete beta functions can be used here. Instead the quantile is found numerically using a derivative free method (__root_finding_TOMS748).]] [[quantile from the complement][Found numerically as above.]] [[mean][ `p * n` ]] [[variance][ `p * n * (1-p)` ]] [[mode][`floor(p * (n + 1))`]] [[skewness][`(1 - 2 * p) / sqrt(n * p * (1 - p))`]] [[kurtosis][`3 - (6 / n) + (1 / (n * p * (1 - p)))`]] [[kurtosis excess][`(1 - 6 * p * q) / (n * p * q)`]] [[parameter estimation][The member functions `find_upper_bound_on_p` `find_lower_bound_on_p` and `find_number_of_trials` are implemented in terms of the inverse incomplete beta functions __ibetac_inv, __ibeta_inv, and __ibetac_invb respectively]] ] [h4 References] * [@http://mathworld.wolfram.com/BinomialDistribution.html Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource]. * [@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia binomial distribution]. * [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm NIST Exploratory Data Analysis]. [endsect] [/section:binomial_dist Binomial] [/ binomial.qbk Copyright 2006 John Maddock and Paul A. Bristow. Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt). ]