1 [section:neg_binom_eg Negative Binomial Distribution Examples]
3 (See also the reference documentation for the __negative_binomial_distrib.)
5 [section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution]
7 Imagine you have a process that follows a negative binomial distribution:
8 for each trial conducted, an event either occurs or does it does not, referred
9 to as "successes" and "failures". The frequency with which successes occur
10 is variously referred to as the
11 success fraction, success ratio, success percentage, occurrence frequency, or probability of occurrence.
13 If, by experiment, you want to measure the
14 the best estimate of success fraction is given simply
15 by /k/ \/ /N/, for /k/ successes out of /N/ trials.
17 However our confidence in that estimate will be shaped by how many trials were conducted,
18 and how many successes were observed. The static member functions
19 `negative_binomial_distribution<>::find_lower_bound_on_p` and
20 `negative_binomial_distribution<>::find_upper_bound_on_p`
21 allow you to calculate the confidence intervals for your estimate of the success fraction.
23 The sample program [@../../example/neg_binom_confidence_limits.cpp
24 neg_binom_confidence_limits.cpp] illustrates their use.
26 [import ../../example/neg_binom_confidence_limits.cpp]
28 [neg_binomial_confidence_limits]
29 Let's see some sample output for a 1 in 10
30 success ratio, first for a mere 20 trials:
32 [pre'''______________________________________________
33 2-Sided Confidence Limits For Success Fraction
34 ______________________________________________
36 Number of successes = 2
37 Number of failures = 18
38 Observed frequency of occurrence = 0.1
39 ___________________________________________
40 Confidence Lower Upper
42 ___________________________________________
43 50.000 0.04812 0.13554
44 75.000 0.03078 0.17727
45 90.000 0.01807 0.22637
46 95.000 0.01235 0.26028
47 99.000 0.00530 0.33111
48 99.900 0.00164 0.41802
49 99.990 0.00051 0.49202
50 99.999 0.00016 0.55574
53 As you can see, even at the 95% confidence level the bounds (0.012 to 0.26) are
54 really very wide, and very asymmetric about the observed value 0.1.
56 Compare that with the program output for a mass
59 [pre'''______________________________________________
60 2-Sided Confidence Limits For Success Fraction
61 ______________________________________________
62 Number of trials = 2000
63 Number of successes = 200
64 Number of failures = 1800
65 Observed frequency of occurrence = 0.1
66 ___________________________________________
67 Confidence Lower Upper
69 ___________________________________________
70 50.000 0.09536 0.10445
71 75.000 0.09228 0.10776
72 90.000 0.08916 0.11125
73 95.000 0.08720 0.11352
74 99.000 0.08344 0.11802
75 99.900 0.07921 0.12336
76 99.990 0.07577 0.12795
77 99.999 0.07282 0.13206
80 Now even when the confidence level is very high, the limits (at 99.999%, 0.07 to 0.13) are really
81 quite close and nearly symmetric to the observed value of 0.1.
83 [endsect][/section:neg_binom_conf Calculating Confidence Limits on the Frequency of Occurrence]
85 [section:neg_binom_size_eg Estimating Sample Sizes for the Negative Binomial.]
87 Imagine you have an event
88 (let's call it a "failure" - though we could equally well call it a success if we felt it was a 'good' event)
89 that you know will occur in 1 in N trials. You may want to know how many trials you need to
90 conduct to be P% sure of observing at least k such failures.
91 If the failure events follow a negative binomial
92 distribution (each trial either succeeds or fails)
93 then the static member function `negative_binomial_distibution<>::find_minimum_number_of_trials`
94 can be used to estimate the minimum number of trials required to be P% sure
95 of observing the desired number of failures.
98 [@../../example/neg_binomial_sample_sizes.cpp neg_binomial_sample_sizes.cpp]
99 demonstrates its usage.
101 [import ../../example/neg_binomial_sample_sizes.cpp]
102 [neg_binomial_sample_sizes]
104 [note Since we're calculating the /minimum/ number of trials required,
105 we'll err on the safe side and take the ceiling of the result.
106 Had we been calculating the
107 /maximum/ number of trials permitted to observe less than a certain
108 number of /failures/ then we would have taken the floor instead. We
109 would also have called `find_minimum_number_of_trials` like this:
111 floor(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
113 which would give us the largest number of trials we could conduct and
114 still be P% sure of observing /failures or less/ failure events, when the
115 probability of success is /p/.]
117 We'll finish off by looking at some sample output, firstly suppose
118 we wish to observe at least 5 "failures" with a 50/50 (0.5) chance of
122 '''Target number of failures = 5, Success fraction = 50%
124 ____________________________
125 Confidence Min Number
127 ____________________________
139 So 18 trials or more would yield a 95% chance that at least our 5
140 required failures would be observed.
142 Compare that to what happens if the success ratio is 90%:
144 [pre'''Target number of failures = 5.000, Success fraction = 90.000%
146 ____________________________
147 Confidence Min Number
149 ____________________________
160 So now 103 trials are required to observe at least 5 failures with
163 [endsect] [/section:neg_binom_size_eg Estimating Sample Sizes.]
165 [section:negative_binomial_example1 Negative Binomial Sales Quota Example.]
168 [@../../example/negative_binomial_example1.cpp negative_binomial_example1.cpp (full source code)]
169 demonstrates a simple use to find the probability of meeting a sales quota.
171 [import ../../example/negative_binomial_example1.cpp]
172 [negative_binomial_eg1_1]
173 [negative_binomial_eg1_2]
175 [endsect] [/section:negative_binomial_example1]
177 [section:negative_binomial_example2 Negative Binomial Table Printing Example.]
178 Example program showing output of a table of values of cdf and pdf for various k failures.
179 [import ../../example/negative_binomial_example2.cpp]
180 [neg_binomial_example2]
181 [neg_binomial_example2_1]
182 [endsect] [/section:negative_binomial_example1 Negative Binomial example 2.]
184 [endsect] [/section:neg_binom_eg Negative Binomial Distribution Examples]
187 Copyright 2006 John Maddock and Paul A. Bristow.
188 Distributed under the Boost Software License, Version 1.0.
189 (See accompanying file LICENSE_1_0.txt or copy at
190 http://www.boost.org/LICENSE_1_0.txt).