Discrete Distributions
Discrete distributions
We start our discussion with the discrete distributions available in AIMMS. They are
the
Binomial
distribution,the
HyperGeometric
distribution,the
Poisson
distribution,the
NegativeBinomial
distribution, andthe
Geometric
distribution.
Discrete distributions describing successes
The Binomial
, HyperGeometric
and Poisson
distributions
describe the number of times that a particular outcome (referred to as
“success”) occurs. In the Binomial
distribution, the underlying
assumption is a fixed number of trials and a constant likelihood of
success. In the HyperGeometric
distribution, the underlying
assumption is “sampling without replacement”: A fixed number of trials
are taken from a population. Each element of this population denotes a
success or failure and cannot occur more than once. In the Poisson
distribution the number of trials is not fixed. Instead we assume that
successes occur independently of each other and with equal chance for
all intervals with the same duration.
Distributions describing trials
The NegativeBinomial
distribution describes the number of failures
before a specified number of successes have occurred. It assumes a
constant chance of success for each trial, so it is linked to the
Binomial
distribution. Similarly, the distribution linked to
Poisson
distribution that describes the amount of time until a
certain number of successes have occurred is known as the Gamma
distribution and is discussed in Continuous Distributions. The
NegativeBinomial
distribution is a special case of the Geometric
distribution and describes the number of failures before the first
success occurs. Similarly, the Exponential
distribution is a special
case of the Gamma
distribution and describes the amount of time
until the first occurrence.
Discrete distributions overview
this table shows the relation between the
discrete distributions. The continuous Exponential
and Gamma
distribution naturally fit in this table as they represent the
distribution of the time it takes before the first/\(n\)th
occurrence (given the average time between two consecutive occurrences).
with replacement 
without replacement 
independent occurrences at random moments 


example 
throwing dice 
drawing cards 
serving customers 
# trials until first success / time until first occurrence 
not supported in AIMMS 


# trials until \(n\)th success / time until \(n\)th occurrence 
not supported in AIMMS 


# successes in fixed # trials / # successes in fixed time 
Binomial
distribution
The Binomial
\((p,n)\) distribution:
Input parameters 
Probability of success \(p\) and number of trials \(n\) 
Input check 
\(\mbox{integer} \; n > 0 \; \mbox{and} \; 0 < p < 1\) 
Permitted values 
\(\{i \;  \; i = 0, 1, \ldots, n \}\) 
Formula 
\(P(X=i) = \binom{n}{i} p^i (1p)^{ni}\) 
Mean 
\(np\) 
Variance 
\(np(1p)\) 
Remarks 

A typical example for this distribution is the number of defectives in a batch of manufactured products where a fixed percentage was found to be defective in previously produced batches. Another example is the number of persons in a group voting yes instead of no, where the probability of yes has been determined on the basis of a sample.
HyperGeometric
distribution
The HyperGeometric
\((p,n,N)\) distribution:
Input parameters 
Known initial probability of success \(p\), number of trials \(n\) and population size \(N\) 
Input check 
\(\mbox{integer} \; n,N: 0 < n \leq N, \; \mbox{and $p \in \frac{1}{N}, \frac{2}{N}, \ldots, \frac{N  1}{N}$}\) 
Permitted values 
\(\{ i \;  \; i = 0, 1, \ldots, n \}\) 
Formula 
\(P(X = i) = \frac{ \binom{N p}{i} \binom{N (1p)}{n  i} }{ \binom{N}{n} }\) 
Mean 
\(np\) 
Variance 
\(np(1p)\mbox{$\frac{Nn}{N1}$}\) 
As an example of this distribution, consider a set of 1000 books of
which 30 are faulty When considering an order containing 50 books from
this set, the HyperGeometric
(0.03,50,1000) distribution shows the
probability of observing \(i \; (i = 0, 1, \ldots,n)\) faulty books
in this subset.
Poisson
distribution
The Poisson
\((\lambda)\) distribution:
Input parameters 
Average number of occurrences \(\lambda\) 
Input check 
\(\lambda > 0\) 
Permitted values 
\(\{ i \;  \; i = 0, 1, \ldots \}\) 
Formula 
\(P(X = i) = \frac{\lambda^i}{i!} e^{\lambda}\) 
Mean 
\(\lambda\) 
Variance 
\(\lambda\) 
Remarks 

The Poisson
distribution should be used when there is an constant
chance of a ‘success’ over time or (as an approximation) when there are
many occurrences with a very small individual chance of ‘success’.
Typical examples are the number of visitors in a day, the number of
errors in a document, the number of defects in a large batch, the number
of telephone calls in a minute, etc.
NegativeBinomial
distribution
The NegativeBinomial
\((p,r)\) distribution:
Input parameters 
Success probability \(p\) and number of successes \(r\) 
Input check 
\(0 < p < 1 \; \mbox{and} \; r = 1, 2, \ldots\) 
Permitted values 
\(\{ i \;  \; i = 0, 1, \ldots \}\) 
Formula 
\(P(X = i) = \binom{r + i  1}{i} p^r (1p)^i\) 
Mean 
\(r/pr\) 
Variance 
\(r(1p)/p^2\) 
Whenever there is a repetition of the same activity, and you are
interested in observing the \(r\)th occurrence of a particular
outcome, then the NegativeBinomial
distribution might be applicable.
A typical situation is going from doortodoor until you have made
\(r\) sales, where the probability of making a sale has been
determined on the basis of previous experience. Note that the
NegativeBinomial
distribution describes the number of failures
before the \(r\)th success. The distribution of the number of
trials \(i\) before the \(r\)th success is given by
\(P_{\texttt{NegativeBinomial(p,r)}}(X=ir)\).
Geometric
distribution
The Geometric
\((p)\) distribution:
Input parameters 
Probability of success \(p\) 
Input check 
\(0 < p < 1\) 
Permitted values 
\(\{i \;  \; i = 0, 1, \ldots \}\) 
Formula 
\(P(X = i) = (1  p)^i p\) 
Mean 
\(1/p1\) 
Variance 
\((1p)/p^2\) 
Remarks 

The Geometric
distribution is a special case of the
NegativeBinomial
distribution. So it can be used for the same type
of problems (the number of visited doors before the first sale). Another
example is an oil company drilling wells until a producing well is
found, where the probability of success is based on measurements around
the site and comparing them with measurements from other similar sites.