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351 lines
13 KiB
Plaintext
351 lines
13 KiB
Plaintext
[section:geometric_dist Geometric Distribution]
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``#include <boost/math/distributions/geometric.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class geometric_distribution;
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typedef geometric_distribution<> geometric;
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template <class RealType, class ``__Policy``>
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class geometric_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor from success_fraction:
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geometric_distribution(RealType p);
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// Parameter accessors:
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RealType success_fraction() const;
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RealType successes() const;
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// Bounds on success fraction:
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static RealType find_lower_bound_on_p(
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RealType trials,
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RealType successes,
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RealType probability); // alpha
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static RealType find_upper_bound_on_p(
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RealType trials,
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RealType successes,
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RealType probability); // alpha
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// Estimate min/max number of trials:
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static RealType find_minimum_number_of_trials(
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RealType k, // Number of failures.
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RealType p, // Success fraction.
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RealType probability); // Probability threshold alpha.
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static RealType find_maximum_number_of_trials(
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RealType k, // Number of failures.
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RealType p, // Success fraction.
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RealType probability); // Probability threshold alpha.
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};
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}} // namespaces
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The class type `geometric_distribution` represents a
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[@http://en.wikipedia.org/wiki/geometric_distribution geometric distribution]:
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it is used when there are exactly two mutually exclusive outcomes of a
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[@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trial]:
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these outcomes are labelled "success" and "failure".
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For [@http://en.wikipedia.org/wiki/Bernoulli_trial Bernoulli trials]
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each with success fraction /p/, the geometric distribution gives
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the probability of observing /k/ trials (failures, events, occurrences, or arrivals)
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before the first success.
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[note For this implementation, the set of trials *includes zero*
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(unlike another definition where the set of trials starts at one, sometimes named /shifted/).]
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The geometric distribution assumes that success_fraction /p/ is fixed for all /k/ trials.
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The probability that there are /k/ failures before the first success is
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__spaces Pr(Y=/k/) = (1-/p/)[super /k/]/p/
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For example, when throwing a 6-face dice the success probability /p/ = 1/6 = 0.1666[recur][space].
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Throwing repeatedly until a /three/ appears,
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the probability distribution of the number of times /not-a-three/ is thrown
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is geometric.
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Geometric distribution has the Probability Density Function PDF:
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__spaces (1-/p/)[super /k/]/p/
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The following graph illustrates how the PDF and CDF vary for three examples
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of the success fraction /p/,
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(when considering the geometric distribution as a continuous function),
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[graph geometric_pdf_2]
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[graph geometric_cdf_2]
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and as discrete.
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[graph geometric_pdf_discrete]
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[graph geometric_cdf_discrete]
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[h4 Related Distributions]
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The geometric distribution is a special case of
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the __negative_binomial_distrib with successes parameter /r/ = 1,
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so only one first and only success is required : thus by definition
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__spaces `geometric(p) == negative_binomial(1, p)`
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negative_binomial_distribution(RealType r, RealType success_fraction);
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negative_binomial nb(1, success_fraction);
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geometric g(success_fraction);
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ASSERT(pdf(nb, 1) == pdf(g, 1));
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This implementation uses real numbers for the computation throughout
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(because it uses the *real-valued* power and exponential functions).
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So to obtain a conventional strictly-discrete geometric distribution
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you must ensure that an integer value is provided for the number of trials
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(random variable) /k/,
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and take integer values (floor or ceil functions) from functions that return
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a number of successes.
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[discrete_quantile_warning geometric]
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[h4 Member Functions]
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[h5 Constructor]
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geometric_distribution(RealType p);
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Constructor: /p/ or success_fraction is the probability of success of a single trial.
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Requires: `0 <= p <= 1`.
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[h5 Accessors]
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RealType success_fraction() const; // successes / trials (0 <= p <= 1)
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Returns the success_fraction parameter /p/ from which this distribution was constructed.
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RealType successes() const; // required successes always one,
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// included for compatibility with negative binomial distribution
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// with successes r == 1.
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Returns unity.
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The following functions are equivalent to those provided for the negative binomial,
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with successes = 1, but are provided here for completeness.
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The best method of calculation for the following functions is disputed:
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see __binomial_distrib and __negative_binomial_distrib for more discussion.
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[h5 Lower Bound on success_fraction Parameter ['p]]
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static RealType find_lower_bound_on_p(
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RealType failures,
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RealType probability) // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
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Returns a *lower bound* on the success fraction:
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[variablelist
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[[failures][The total number of failures before the 1st success.]]
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[[alpha][The largest acceptable probability that the true value of
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the success fraction is [*less than] the value returned.]]
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]
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For example, if you observe /k/ failures from /n/ trials
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the best estimate for the success fraction is simply 1/['n], but if you
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want to be 95% sure that the true value is [*greater than] some value,
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['p[sub min]], then:
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p``[sub min]`` = geometric_distribution<RealType>::
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find_lower_bound_on_p(failures, 0.05);
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative_binomial confidence interval example.]
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This function uses the Clopper-Pearson method of computing the lower bound on the
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success fraction, whilst many texts refer to this method as giving an "exact"
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result in practice it produces an interval that guarantees ['at least] the
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coverage required, and may produce pessimistic estimates for some combinations
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of /failures/ and /successes/. See:
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[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
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Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].
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[h5 Upper Bound on success_fraction Parameter p]
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static RealType find_upper_bound_on_p(
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RealType trials,
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RealType alpha); // (0 <= alpha <= 1), 0.05 equivalent to 95% confidence.
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Returns an *upper bound* on the success fraction:
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[variablelist
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[[trials][The total number of trials conducted.]]
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[[alpha][The largest acceptable probability that the true value of
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the success fraction is [*greater than] the value returned.]]
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]
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For example, if you observe /k/ successes from /n/ trials the
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best estimate for the success fraction is simply ['k/n], but if you
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want to be 95% sure that the true value is [*less than] some value,
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['p[sub max]], then:
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p``[sub max]`` = geometric_distribution<RealType>::find_upper_bound_on_p(
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k, 0.05);
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_conf See negative binomial confidence interval example.]
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This function uses the Clopper-Pearson method of computing the lower bound on the
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success fraction, whilst many texts refer to this method as giving an "exact"
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result in practice it produces an interval that guarantees ['at least] the
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coverage required, and may produce pessimistic estimates for some combinations
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of /failures/ and /successes/. See:
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[@http://www.ucs.louisiana.edu/~kxk4695/Discrete_new.pdf
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Yong Cai and K. Krishnamoorthy, A Simple Improved Inferential Method for Some Discrete Distributions.
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Computational statistics and data analysis, 2005, vol. 48, no3, 605-621].
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[h5 Estimating Number of Trials to Ensure at Least a Certain Number of Failures]
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static RealType find_minimum_number_of_trials(
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RealType k, // number of failures.
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RealType p, // success fraction.
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RealType alpha); // probability threshold (0.05 equivalent to 95%).
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This functions estimates the number of trials required to achieve a certain
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probability that [*more than ['k] failures will be observed].
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[variablelist
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[[k][The target number of failures to be observed.]]
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[[p][The probability of ['success] for each trial.]]
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[[alpha][The maximum acceptable ['risk] that only ['k] failures or fewer will be observed.]]
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]
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For example:
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geometric_distribution<RealType>::find_minimum_number_of_trials(10, 0.5, 0.05);
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Returns the smallest number of trials we must conduct to be 95% (1-0.05) sure
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of seeing 10 failures that occur with frequency one half.
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[link math_toolkit.stat_tut.weg.neg_binom_eg.neg_binom_size_eg Worked Example.]
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This function uses numeric inversion of the geometric distribution
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to obtain the result: another interpretation of the result is that it finds
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the number of trials (failures) that will lead to an /alpha/ probability
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of observing /k/ failures or fewer.
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[h5 Estimating Number of Trials to Ensure a Maximum Number of Failures or Less]
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static RealType find_maximum_number_of_trials(
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RealType k, // number of failures.
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RealType p, // success fraction.
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RealType alpha); // probability threshold (0.05 equivalent to 95%).
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This functions estimates the maximum number of trials we can conduct and achieve
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a certain probability that [*k failures or fewer will be observed].
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[variablelist
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[[k][The maximum number of failures to be observed.]]
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[[p][The probability of ['success] for each trial.]]
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[[alpha][The maximum acceptable ['risk] that more than ['k] failures will be observed.]]
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]
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For example:
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geometric_distribution<RealType>::find_maximum_number_of_trials(0, 1.0-1.0/1000000, 0.05);
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Returns the largest number of trials we can conduct and still be 95% sure
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of seeing no failures that occur with frequency one in one million.
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This function uses numeric inversion of the geometric distribution
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to obtain the result: another interpretation of the result, is that it finds
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the number of trials that will lead to an /alpha/ probability
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of observing more than k failures.
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[h4 Non-member Accessors]
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All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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that are generic to all distributions are supported: __usual_accessors.
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However it's worth taking a moment to define what these actually mean in
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the context of this distribution:
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[table Meaning of the non-member accessors.
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[[Function][Meaning]]
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[[__pdf]
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[The probability of obtaining [*exactly k failures] from /k/ trials
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with success fraction p. For example:
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``pdf(geometric(p), k)``]]
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[[__cdf]
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[The probability of obtaining [*k failures or fewer] from /k/ trials
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with success fraction p and success on the last trial. For example:
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``cdf(geometric(p), k)``]]
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[[__ccdf]
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[The probability of obtaining [*more than k failures] from /k/ trials
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with success fraction p and success on the last trial. For example:
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``cdf(complement(geometric(p), k))``]]
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[[__quantile]
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[The [*greatest] number of failures /k/ expected to be observed from /k/ trials
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with success fraction /p/, at probability /P/. Note that the value returned
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is a real-number, and not an integer. Depending on the use case you may
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want to take either the floor or ceiling of the real result. For example:
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``quantile(geometric(p), P)``]]
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[[__quantile_c]
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[The [*smallest] number of failures /k/ expected to be observed from /k/ trials
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with success fraction /p/, at probability /P/. Note that the value returned
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is a real-number, and not an integer. Depending on the use case you may
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want to take either the floor or ceiling of the real result. For example:
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``quantile(complement(geometric(p), P))``]]
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]
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[h4 Accuracy]
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This distribution is implemented using the pow and exp functions, so most results
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are accurate within a few epsilon for the RealType.
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For extreme values of `double` /p/, for example 0.9999999999,
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accuracy can fall significantly, for example to 10 decimal digits (from 16).
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[h4 Implementation]
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In the following table, /p/ is the probability that any one trial will
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be successful (the success fraction), /k/ is the number of failures,
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/p/ is the probability and /q = 1-p/,
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/x/ is the given probability to estimate
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the expected number of failures using the quantile.
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[table
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[[Function][Implementation Notes]]
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[[pdf][pdf = p * pow(q, k)]]
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[[cdf][cdf = 1 - q[super k=1]]]
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[[cdf complement][exp(log1p(-p) * (k+1))]]
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[[quantile][k = log1p(-x) / log1p(-p) -1]]
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[[quantile from the complement][k = log(x) / log1p(-p) -1]]
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[[mean][(1-p)/p]]
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[[variance][(1-p)/p[sup2]]]
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[[mode][0]]
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[[skewness][(2-p)/[sqrt]q]]
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[[kurtosis][9+p[sup2]/q]]
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[[kurtosis excess][6 +p[sup2]/q]]
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[[parameter estimation member functions][See __negative_binomial_distrib]]
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[[`find_lower_bound_on_p`][See __negative_binomial_distrib]]
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[[`find_upper_bound_on_p`][See __negative_binomial_distrib]]
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[[`find_minimum_number_of_trials`][See __negative_binomial_distrib]]
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[[`find_maximum_number_of_trials`][See __negative_binomial_distrib]]
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]
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[endsect][/section:geometric_dist geometric]
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[/ geometric.qbk
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Copyright 2010 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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