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Geometric Distribution Examples

For this example, we will opt to #define two macros to control the error and discrete handling policies. For this simple example, we want to avoid throwing an exception (the default policy) and just return infinity. We want to treat the distribution as if it was continuous, so we choose a discrete_quantile policy of real, rather than the default policy integer_round_outwards.


[Caution] Caution

It is vital to #include distributions etc after the above #defines

After that we need some includes to provide easy access to the negative binomial distribution, and we need some std library iostream, of course.

#include <boost/math/distributions/geometric.hpp>
  // for geometric_distribution
  using ::boost::math::geometric_distribution; // 
  using ::boost::math::geometric; // typedef provides default type is double.
  using  ::boost::math::pdf; // Probability mass function.
  using  ::boost::math::cdf; // Cumulative density function.
  using  ::boost::math::quantile;

#include <boost/math/distributions/negative_binomial.hpp>
  // for negative_binomial_distribution
  using boost::math::negative_binomial; // typedef provides default type is double.

#include <boost/math/distributions/normal.hpp>
  // for negative_binomial_distribution
  using boost::math::normal; // typedef provides default type is double.

#include <iostream>
  using std::cout; using std::endl;
  using std::noshowpoint; using std::fixed; using std::right; using std::left;
#include <iomanip>
  using std::setprecision; using std::setw;

#include <limits>
  using std::numeric_limits;

It is always sensible to use try and catch blocks because defaults policies are to throw an exception if anything goes wrong.

Simple try'n'catch blocks (see below) will ensure that you get a helpful error message instead of an abrupt (and silent) program abort.

Throwing a dice

The Geometric distribution describes the probability (p) of a number of failures to get the first success in k Bernoulli trials. (A Bernoulli trial is one with only two possible outcomes, success of failure, and p is the probability of success).

Suppose an 'fair' 6-face dice is thrown repeatedly:

double success_fraction = 1./6; // success_fraction (p) = 0.1666
// (so failure_fraction is 1 - success_fraction = 5./6 = 1- 0.1666 = 0.8333)

If the dice is thrown repeatedly until the first time a three appears. The probablility distribution of the number of times it is thrown not getting a three (not-a-threes number of failures to get a three) is a geometric distribution with the success_fraction = 1/6 = 0.1666 ̇.

We therefore start by constructing a geometric distribution with the one parameter success_fraction, the probability of success.

geometric g6(success_fraction); // type double by default.

To confirm, we can echo the success_fraction parameter of the distribution.

cout << "success fraction of a six-sided dice is " << g6.success_fraction() << endl;

So the probability of getting a three at the first throw (zero failures) is

cout << pdf(g6, 0) << endl; // 0.1667
cout << cdf(g6, 0) << endl; // 0.1667

Note that the cdf and pdf are identical because the is only one throw. If we want the probability of getting the first three on the 2nd throw:

cout << pdf(g6, 1) << endl; // 0.1389

If we want the probability of getting the first three on the 1st or 2nd throw (allowing one failure):

cout << "pdf(g6, 0) + pdf(g6, 1) = " << pdf(g6, 0) + pdf(g6, 1) << endl;

Or more conveniently, and more generally, we can use the Cumulative Distribution Function CDF.

cout << "cdf(g6, 1) = " << cdf(g6, 1) << endl; // 0.3056

If we allow many more (12) throws, the probability of getting our three gets very high:

cout << "cdf(g6, 12) = " << cdf(g6, 12) << endl; // 0.9065 or 90% probability.

If we want to be much more confident, say 99%, we can estimate the number of throws to be this sure using the inverse or quantile.

cout << "quantile(g6, 0.99) = " << quantile(g6, 0.99) << endl; // 24.26

Note that the value returned is not an integer: if you want an integer result you should use either floor, round or ceil functions, or use the policies mechanism. See Understanding Quantiles of Discrete Distributions

The geometric distribution is related to the negative binomial    negative_binomial_distribution(RealType r, RealType p); with parameter r = 1. So we could get the same result using the negative binomial, but using the geometric the results will be faster, and may be more accurate.

negative_binomial nb(1, success_fraction);
cout << pdf(nb, 1) << endl; // 0.1389
cout << cdf(nb, 1) << endl; // 0.3056

We could also the complement to express the required probability as 1 - 0.99 = 0.01 (and get the same result):

cout << "quantile(complement(g6, 1 - p))  " << quantile(complement(g6, 0.01)) << endl; // 24.26

Note too that Boost.Math geometric distribution is implemented as a continuous function. Unlike other implementations (for example R) it uses the number of failures as a real parameter, not as an integer. If you want this integer behaviour, you may need to enforce this by rounding the parameter you pass, probably rounding down, to the nearest integer. For example, R returns the success fraction probability for all values of failures from 0 to 0.999999 thus:

   R> formatC(pgeom(0.0001,0.5, FALSE), digits=17) "               0.5"

So in Boost.Math the equivalent is

    geometric g05(0.5);  // Probability of success = 0.5 or 50%
    // Output all potentially significant digits for the type, here double.

  int max_digits10 = 2 + (boost::math::policies::digits<double, boost::math::policies::policy<> >() * 30103UL) / 100000UL;
  cout << "BOOST_NO_NUMERIC_LIMITS_LOWEST is defined" << endl;
  int max_digits10 = std::numeric_limits<double>::max_digits10;
  cout << "Show all potentially significant decimal digits std::numeric_limits<double>::max_digits10 = "
    << max_digits10 << endl;
  cout.precision(max_digits10); // 

    cout << cdf(g05, 0.0001) << endl; // returns 0.5000346561579232, not exact 0.5.

To get the R discrete behaviour, you simply need to round with, for example, the floor function.

cout << cdf(g05, floor(0.0001)) << endl; // returns exactly 0.5

> formatC(pgeom(0.9999999,0.5, FALSE), digits=17) [1] "              0.25"
> formatC(pgeom(1.999999,0.5, FALSE), digits=17)[1] "              0.25" k = 1
> formatC(pgeom(1.9999999,0.5, FALSE), digits=17)[1] "0.12500000000000003" k = 2

shows that R makes an arbitrary round-up decision at about 1e7 from the next integer above. This may be convenient in practice, and could be replicated in C++ if desired.

Surveying customers to find one with a faulty product

A company knows from warranty claims that 2% of their products will be faulty, so the 'success_fraction' of finding a fault is 0.02. It wants to interview a purchaser of faulty products to assess their 'user experience'.

To estimate how many customers they will probably need to contact in order to find one who has suffered from the fault, we first construct a geometric distribution with probability 0.02, and then chose a confidence, say 80%, 95%, or 99% to finding a customer with a fault. Finally, we probably want to round up the result to the integer above using the ceil function. (We could also use a policy, but that is hardly worthwhile for this simple application.)

(This also assumes that each customer only buys one product: if customers bought more than one item, the probability of finding a customer with a fault obviously improves.)

geometric g(0.02); // On average, 2 in 100 products are faulty.
double c = 0.95; // 95% confidence.
cout << " quantile(g, " << c << ") = " << quantile(g, c) << endl;

cout << "To be " << c * 100
  << "% confident of finding we customer with a fault, need to survey "
  <<  ceil(quantile(g, c)) << " customers." << endl; // 148
c = 0.99; // Very confident.
cout << "To be " << c * 100
  << "% confident of finding we customer with a fault, need to survey "
  <<  ceil(quantile(g, c)) << " customers." << endl; // 227
c = 0.80; // Only reasonably confident.
cout << "To be " << c * 100
  << "% confident of finding we customer with a fault, need to survey "
  <<  ceil(quantile(g, c)) << " customers." << endl; // 79

Basket Ball Shooters

According to Wikipedia, average pro basket ball players get free throws in the baskets 70 to 80 % of the time, but some get as high as 95%, and others as low as 50%. Suppose we want to compare the probabilities of failing to get a score only on the first or on the fifth shot? To start we will consider the average shooter, say 75%. So we construct a geometric distribution with success_fraction parameter 75/100 = 0.75.

geometric gav(0.75); // Shooter averages 7.5 out of 10 in the basket.

What is probability of getting 1st try in the basket, that is with no failures?

cout << "Probability of score on 1st try = " << pdf(gav, 0) << endl; // 0.75

This is, of course, the success_fraction probability 75%. What is the probability that the shooter only scores on the fifth shot? So there are 5-1 = 4 failures before the first success.

cout << "Probability of score on 5th try = " << pdf(gav, 4) << endl; // 0.0029

Now compare this with the poor and the best players success fraction. We need to constructing new distributions with the different success fractions, and then get the corresponding probability density functions values:

geometric gbest(0.95);
cout << "Probability of score on 5th try = " << pdf(gbest, 4) << endl; // 5.9e-6
geometric gmediocre(0.50);
cout << "Probability of score on 5th try = " << pdf(gmediocre, 4) << endl; // 0.031

So we can see the very much smaller chance (0.000006) of 4 failures by the best shooters, compared to the 0.03 of the mediocre.

Estimating failures

Of course one man's failure is an other man's success. So a fault can be defined as a 'success'.

If a fault occurs once after 100 flights, then one might naively say that the risk of fault is obviously 1 in 100 = 1/100, a probability of 0.01.

This is the best estimate we can make, but while it is the truth, it is not the whole truth, for it hides the big uncertainty when estimating from a single event. "One swallow doesn't make a summer." To show the magnitude of the uncertainty, the geometric (or the negative binomial) distribution can be used.

If we chose the popular 95% confidence in the limits, corresponding to an alpha of 0.05, because we are calculating a two-sided interval, we must divide alpha by two.

double alpha = 0.05;
double k = 100; // So frequency of occurence is 1/100.
cout << "Probability is failure is " << 1/k << endl;
double t = geometric::find_lower_bound_on_p(k, alpha/2);
cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
  << t << endl; // 0.00025
t = geometric::find_upper_bound_on_p(k, alpha/2);
cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
  << t << endl; // 0.037

So while we estimate the probability is 0.01, it might lie between 0.0003 and 0.04. Even if we relax our confidence to alpha = 90%, the bounds only contract to 0.0005 and 0.03. And if we require a high confidence, they widen to 0.00005 to 0.05.

alpha = 0.1; // 90% confidence.
t = geometric::find_lower_bound_on_p(k, alpha/2);
cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
  << t << endl; // 0.0005
t = geometric::find_upper_bound_on_p(k, alpha/2);
cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
  << t << endl; // 0.03

alpha = 0.01; // 99% confidence.
t = geometric::find_lower_bound_on_p(k, alpha/2);
cout << "geometric::find_lower_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
  << t << endl; // 5e-005
t = geometric::find_upper_bound_on_p(k, alpha/2);
cout << "geometric::find_upper_bound_on_p(" << int(k) << ", " << alpha/2 << ") = "
    << t << endl; // 0.052

In real life, there will usually be more than one event (fault or success), when the negative binomial, which has the neccessary extra parameter, will be needed.

As noted above, using a catch block is always a good idea, even if you hope not to use it!

catch(const std::exception& e)
{ // Since we have set an overflow policy of ignore_error,
  // an overflow exception should never be thrown.
   std::cout << "\nMessage from thrown exception was:\n " << e.what() << std::endl;

For example, without a ignore domain error policy, if we asked for

pdf(g, -1)

for example, we would get an unhelpful abort, but with a catch:

Message from thrown exception was:
 Error in function boost::math::pdf(const exponential_distribution<double>&, double):
 Number of failures argument is -1, but must be >= 0 !

See full source C++ of this example at geometric_examples.cpp

See negative_binomial confidence interval example.