
BoostCommit : 
From: pbristow_at_[hidden]
Date: 20070921 07:46:06
Author: pbristow
Date: 20070921 07:46:06 EDT (Fri, 21 Sep 2007)
New Revision: 39439
URL: http://svn.boost.org/trac/boost/changeset/39439
Log:
Source updated, minor changes to match
Text files modified:
sandbox/math_toolkit/libs/math/doc/distributions/negative_binomial_example.qbk  62 +
1 files changed, 3 insertions(+), 59 deletions()
Modified: sandbox/math_toolkit/libs/math/doc/distributions/negative_binomial_example.qbk
==============================================================================
 sandbox/math_toolkit/libs/math/doc/distributions/negative_binomial_example.qbk (original)
+++ sandbox/math_toolkit/libs/math/doc/distributions/negative_binomial_example.qbk 20070921 07:46:06 EDT (Fri, 21 Sep 2007)
@@ 164,69 +164,13 @@
[section:negative_binomial_example1 Negative Binomial Sales Quota Example.]
The example program
+This example program
[@../../example/negative_binomial_example1.cpp negative_binomial_example1.cpp (full source code)]
demonstrates a simple use to find the probability of meeting a sale quota.

Based on [@http://en.wikipedia.org/wiki/Negative_binomial_distribution
a problem by Dr. Diane Evans,
Professor of Mathematics at RoseHulman Institute of Technology].

Pat is required to sell candy bars to raise money for the 6th grade field trip.
There are thirty houses in the neighborhood,
and Pat is not supposed to return home until five candy bars have been sold.
So the child goes door to door, selling candy bars.
At each house, there is a 0.4 probability (40%) of selling one candy bar
and a 0.6 probability (60%) of selling nothing.

What is the probability mass (density) function for selling the last (fifth)
candy bar at the nth house?

The Negative Binomial(r, p) distribution describes the probability of k failures
and r successes in k+r Bernoulli(p) trials with success on the last trial.
(A [@http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli trial]
is one with only two possible outcomes, success of failure,
and p is the probability of success).
Selling five candy bars means getting five successes, so successes r = 5.
The total number of trials (n) in this case, houses visited) this takes is therefore
 = sucesses + failures or k + r = k + 5.
The random variable we are interested in is the number of houses (k)
that must be visited to sell five candy bars,
so we substitute k = n  5 into a negative_binomial(5, 0.4) mass (density) function
and obtain the following mass (density) function of the distribution of houses (for n >= 5):
Obviously, the best case is that Pat makes sales on all the first five houses.

What is the probability that Pat finishes /on the tenth house/?

 f(10) = 0.1003290624, or about 1 in 10

What is the probability that Pat finishes /on or before/ reaching the eighth house?

To finish on or before the eighth house,
Pat must finish at the fifth, sixth, seventh, or eighth house.
Sum those probabilities:

 f(5) = 0.01024
 f(6) = 0.03072
 f(7) = 0.055296
 f(8) = 0.0774144
 sum {j=5 to 8} f(j) = 0.17367

What is the probability that Pat exhausts all 30 houses in the neighborhood,
and still doesn't sell the required 5 candy bars?

1  sum{j=5 to 30} f(j) = 1  incomplete beta (p = 0.4)(5, 305+1) =~ 1  0.99849 = 0.00151 = 0.15%.

See also [@ http://en.wikipedia.org/wiki/Bernoulli_distribution Bernoulli distribution]
and [@http://www.math.uah.edu/stat/bernoulli/Introduction.xhtml Bernoulli applications].

In this example, we will deliberately produce a variety of calculations
and outputs to demonstrate the ways that the negative binomial distribution
can be implemented with this library,
and it is also deliberately overcommented.
+demonstrates a simple use to find the probability of meeting a sales quota.
[import ../../example/negative_binomial_example1.cpp]
[negative_binomial_eg1_1]
+[negative_binomial_eg1_2]
[endsect] [/section:negative_binomial_example1]
BoostCommit list run by bdawes at acm.org, david.abrahams at rcn.com, gregod at cs.rpi.edu, cpdaniel at pacbell.net, john at johnmaddock.co.uk