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250 lines
10 KiB
Plaintext
[section:neg_binom_eg Negative Binomial Distribution Examples]
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(See also the reference documentation for the __negative_binomial_distrib.)
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[section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution]
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Imagine you have a process that follows a negative binomial distribution:
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for each trial conducted, an event either occurs or does it does not, referred
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to as "successes" and "failures". The frequency with which successes occur
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is variously referred to as the
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success fraction, success ratio, success percentage, occurrence frequency, or probability of occurrence.
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If, by experiment, you want to measure the
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the best estimate of success fraction is given simply
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by /k/ \/ /N/, for /k/ successes out of /N/ trials.
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However our confidence in that estimate will be shaped by how many trials were conducted,
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and how many successes were observed. The static member functions
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`negative_binomial_distribution<>::find_lower_bound_on_p` and
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`negative_binomial_distribution<>::find_upper_bound_on_p`
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allow you to calculate the confidence intervals for your estimate of the success fraction.
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The sample program [@../../example/neg_binomial_confidence_limits.cpp
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neg_binomial_confidence_limits.cpp] illustrates their use.
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[import ../../example/neg_binomial_confidence_limits.cpp]
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[neg_binomial_confidence_limits]
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Let's see some sample output for a 1 in 10
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success ratio, first for a mere 20 trials:
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[pre'''______________________________________________
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2-Sided Confidence Limits For Success Fraction
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______________________________________________
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Number of trials = 20
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Number of successes = 2
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Number of failures = 18
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Observed frequency of occurrence = 0.1
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___________________________________________
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Confidence Lower Upper
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Value (%) Limit Limit
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___________________________________________
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50.000 0.04812 0.13554
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75.000 0.03078 0.17727
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90.000 0.01807 0.22637
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95.000 0.01235 0.26028
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99.000 0.00530 0.33111
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99.900 0.00164 0.41802
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99.990 0.00051 0.49202
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99.999 0.00016 0.55574
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''']
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As you can see, even at the 95% confidence level the bounds (0.012 to 0.26) are
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really very wide, and very asymmetric about the observed value 0.1.
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Compare that with the program output for a mass
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2000 trials:
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[pre'''______________________________________________
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2-Sided Confidence Limits For Success Fraction
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______________________________________________
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Number of trials = 2000
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Number of successes = 200
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Number of failures = 1800
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Observed frequency of occurrence = 0.1
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___________________________________________
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Confidence Lower Upper
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Value (%) Limit Limit
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___________________________________________
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50.000 0.09536 0.10445
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75.000 0.09228 0.10776
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90.000 0.08916 0.11125
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95.000 0.08720 0.11352
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99.000 0.08344 0.11802
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99.900 0.07921 0.12336
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99.990 0.07577 0.12795
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99.999 0.07282 0.13206
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''']
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Now even when the confidence level is very high, the limits (at 99.999%, 0.07 to 0.13) are really
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quite close and nearly symmetric to the observed value of 0.1.
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[endsect][/section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence]
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[section:neg_binom_size_eg Estimating Sample Sizes for the Negative Binomial.]
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Imagine you have an event
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(let's call it a "failure" - though we could equally well call it a success if we felt it was a 'good' event)
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that you know will occur in 1 in N trials. You may want to know how many trials you need to
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conduct to be P% sure of observing at least k such failures.
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If the failure events follow a negative binomial
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distribution (each trial either succeeds or fails)
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then the static member function `negative_binomial_distibution<>::find_minimum_number_of_trials`
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can be used to estimate the minimum number of trials required to be P% sure
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of observing the desired number of failures.
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The example program
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[@../../example/neg_binomial_sample_sizes.cpp neg_binomial_sample_sizes.cpp]
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demonstrates its usage.
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[import ../../example/neg_binomial_sample_sizes.cpp]
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[neg_binomial_sample_sizes]
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[note Since we're calculating the /minimum/ number of trials required,
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we'll err on the safe side and take the ceiling of the result.
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Had we been calculating the
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/maximum/ number of trials permitted to observe less than a certain
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number of /failures/ then we would have taken the floor instead. We
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would also have called `find_minimum_number_of_trials` like this:
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``
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floor(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
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``
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which would give us the largest number of trials we could conduct and
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still be P% sure of observing /failures or less/ failure events, when the
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probability of success is /p/.]
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We'll finish off by looking at some sample output, firstly suppose
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we wish to observe at least 5 "failures" with a 50/50 (0.5) chance of
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success or failure:
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[pre
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'''Target number of failures = 5, Success fraction = 50%
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____________________________
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Confidence Min Number
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Value (%) Of Trials
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____________________________
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50.000 11
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75.000 14
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90.000 17
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95.000 18
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99.000 22
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99.900 27
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99.990 31
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99.999 36
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'''
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]
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So 18 trials or more would yield a 95% chance that at least our 5
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required failures would be observed.
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Compare that to what happens if the success ratio is 90%:
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[pre'''Target number of failures = 5.000, Success fraction = 90.000%
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____________________________
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Confidence Min Number
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Value (%) Of Trials
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____________________________
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50.000 57
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75.000 73
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90.000 91
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95.000 103
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99.000 127
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99.900 159
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99.990 189
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99.999 217
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''']
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So now 103 trials are required to observe at least 5 failures with
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95% certainty.
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[endsect] [/section:neg_binom_size_eg Estimating Sample Sizes.]
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[section:negative_binomial_example1 Negative Binomial example 1.]
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The example program
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[@../../example/negative_binomial_example1.cpp negative_binomial_example1.cpp (full source code)]
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demonstrates a simple use to find the probability of meeting a sale quota.
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Based on [@http://en.wikipedia.org/wiki/Negative_binomial_distribution
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a problem by Dr. Diane Evans,
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Professor of Mathematics at Rose-Hulman Institute of Technology].
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Pat is required to sell candy bars to raise money for the 6th grade field trip.
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There are thirty houses in the neighborhood,
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and Pat is not supposed to return home until five candy bars have been sold.
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So the child goes door to door, selling candy bars.
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At each house, there is a 0.4 probability (40%) of selling one candy bar
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and a 0.6 probability (60%) of selling nothing.
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What is the probability mass (density) function for selling the last (fifth)
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candy bar at the nth house?
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The Negative Binomial(r, p) distribution describes the probability of k failures
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and r successes in k+r Bernoulli(p) trials with success on the last trial.
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(A [@http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli trial]
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is one with only two possible outcomes, success of failure,
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and p is the probability of success).
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Selling five candy bars means getting five successes, so successes r = 5.
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The total number of trials (n) in this case, houses visited) this takes is therefore
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= sucesses + failures or k + r = k + 5.
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The random variable we are interested in is the number of houses (k)
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that must be visited to sell five candy bars,
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so we substitute k = n 5 into a negative_binomial(5, 0.4) mass (density) function
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and obtain the following mass (density) function of the distribution of houses (for n >= 5):
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Obviously, the best case is that Pat makes sales on all the first five houses.
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What is the probability that Pat finishes /on the tenth house/?
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f(10) = 0.1003290624, or about 1 in 10
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What is the probability that Pat finishes /on or before/ reaching the eighth house?
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To finish on or before the eighth house,
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Pat must finish at the fifth, sixth, seventh, or eighth house.
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Sum those probabilities:
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f(5) = 0.01024
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f(6) = 0.03072
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f(7) = 0.055296
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f(8) = 0.0774144
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sum {j=5 to 8} f(j) = 0.17367
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What is the probability that Pat exhausts all 30 houses in the neighborhood,
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and still doesn't sell the required 5 candy bars?
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1 - sum{j=5 to 30} f(j) = 1 - incomplete beta (p = 0.4)(5, 30-5+1) =~ 1 - 0.99849 = 0.00151 = 0.15%.
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See also [@ http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution]
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and [@http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml Bernoulli applications].
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In this example, we will deliberately produce a variety of calculations
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and outputs to demonstrate the ways that the negative binomial distribution
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can be implemented with this library,
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and it is also deliberately over-commented.
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[import ../../example/negative_binomial_example1.cpp]
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[negative_binomial_eg1_1]
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[endsect] [/section:negative_binomial_example1]
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[section:negative_binomial_example2 Negative Binomial example 2.]
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Example program showing output of a table of values of cdf and pdf for various k failures.
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[import ../../example/negative_binomial_example2.cpp]
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[neg_binomial_example2]
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[neg_binomial_example2_1]
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[endsect] [/section:negative_binomial_example1 Negative Binomial example 2.]
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[section:negative_binomial_example3 Negative Binomial example 3.]
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The example program
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[@../../example/negative_binomial_example3.cpp negative_binomial_example3.cpp (full source code)]
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demonstrates example from K. Krishnamoorthy.
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[import ../../example/negative_binomial_example3.cpp]
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[neg_binomial_example3]
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[neg_binomial_example3_1]
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[endsect] [/section:negative_binomial_example1 Negative Binomial example 3.]
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[endsect] [/section:neg_binom_eg Negative Binomial Distribution Examples]
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