178 ◾ Simple Statistical Methods for Software Engineering
Review Questions
1. Give an example of the application of binomial distribution.
2. Give an example of the application of the hypergeometric distribution.
3. Give an example of the application of the negative binomial distribution.
4. Give an example of the application of the geometric distribution.
5. What is Bayes theorem?
BOX 11.5 THE THEORY THAT WOULD NOT DIE
Sharon McGrayne’s book, e eory at Would Not Die: How Bayes’ Rule
Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged
Triumphant from Two Centuries of Controversy, presents a history of Bayes’
theorem. e following is an excerpt from the review of this book in http://
www .less wrong.com.
Bayes’ system was Initial Belief + New Data → Improved Belief. Mathe-
maticians were horrified to see something as whimsical as a guess play a role
in rigorous mathematics; this problem of priors was insurmountable.
Pierre-Simon Laplace, a brilliant young mathematician, and the world’s
first Bayesian, came to believe that probability theory held the key, and he
independently rediscovered Bayes’ mechanism.
Joseph Bertrand was convinced that Bayes’ theorem was the only way for
artillery officers to correctly deal with a host of uncertainties about the ene-
mies’ location, air density, wind direction, and more.
Geologist Harold Jeffreys made Bayes’ theorem useful for scientists, pro-
posing it as an alternative to Fisher’s p-values and significance tests, which
depended on “imaginary repetitions.”
For decades, Fisher and Jeffreys were the world’s two greatest statisticians,
traded blows over probability theory in scientific journals and in public. Fisher
was louder and bolder, and frequentism was easier to use than Bayesianism.
is marked a short lived decline of the Bayesian paradigm.
In 1983, the US Air Force sponsored a review of NASA’s estimates of
the probability of shuttle failure. NASA’s estimate was 1 in 100,000. e
contractor used Bayes and estimated the odds of rocket booster failure at
1 in 35. In 1986, Challenger exploded. Frequentist statistics worked okay
when one hypothesis was a special case of another, but when hypotheses
were competing and abrupt changes were in the data, frequentism did not
work.
One challenge had always been that Bayesian statistical operations were
harder to calculate, and computers were still quite slow. is changed in the
1990s, when computers became much faster and cheaper than before.
The Law of Large Numbers ◾ 179
Exercises
1. In a certain school, it has been estimated that the probability of students pass-
ing mathematics tests is 69%. Find out the probability of at least 80 passes
in a batch of 89 students. Plot the related binomial distribution. Use Excel
function BINOM.DIST for your calculations.
2. In a certain application 12 modules have just been built. Test results show that
there are 4 defective modules. If we draw samples of size 3 without replace-
ment, find the probability that a sample contains two defective modules. Use
the Excel function HYPGEOM.DIST for your calculations.
3. Right first-time design probability in a software development project is esti-
mated at 0.3. Estimate the probability of needing seven trials to find a defect-
free feature design. Plot a graph between trials and geometric probability.
References
1. B. Singh, R. Viveros and D. L. Parnas, Estimating Software Reliability Using Inverse Sampl ing,
Communications Research Laboratory, McMaster University, Hamilton Department
of Mathematics and Statistics, e College of Information Sciences and Technology,
e Pennsylvania State University. Available at http://citeseerx.ist.psu.edu/viewdoc
/download?doi=10.1.1.71.1577&rep=rep1&type=pdf.
2. M. Wisal, Formal Verification of Negative Binomial Distribution in Higher Order Logic,
Department of Computer Engineering, University of Engineering and Technology,
Taxila, 2012.
3. D. R. Bellhouse, e Reverend omas Bayes, FRS: A biography to celebrate the ter-
centenary of his birth, Statistical Science, 19(1), 3–43, 2004.
4. T. Lohrbeer, Bayesian Maths for Dummies. Available at http://blog.fastfedora.com
/2010/12/bayesian-math-for-dummies.html.
5. S. Chulani, B. Boehm and B. Steece, Bayesian Analysis of Empirical Software Engineering Cost
Models, University of Southern California, IEEE Transactins on Software Engineering,
USC-CSE, 1999.
6. N. Fenton, Software Project and Quality Modelling Using Bayesian Networks. Available
at http://www.eecs.qmul.ac.uk/~norman/papers/software_project_quality.pdf.
7. S. Bibi, I. Stamelos and L. Angelis, Bayesian Belief Networks as a Software Productivity Esti-
mation Tool, Department of Informatics, Aristotle University, essaloniki, Greece, 2003.
8. S. Wagner, A Bayesian network approach to assess and predict software quality using
activity-based quality models. In: Proceeding PROMISE ‘09 Proceedings of the 5th
International Conference on Predictor Models in Software Engineering, Article No. 6,
ACM, New York, 2009.
Suggested Reading
Wroughton, J. and T. Cole, Distinguishing between binomial, hypergeometric and negative
binomial distributions, Journal of Statistics Education, 21(1), 2013. Available at http://
www.amstat.org/publications/jse/v21n1/wroughton.pdf.
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