Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 54

Table 2.1 Fits of Poisson and negative binomial distributions to the spatial distribution of oil
deposits for 5 5 miles grid areas, Alberta

Deposits Observed frequency Poisson frequency Negative binomial

0 8,586 8,508.5 8,584.3

1 176 303.0 176.8

2 35 5.4 39.1

3 13 0.1 11.3

4 6 0.0 3.6

5 1 0.0 1.2

6 or more 0 0.0 0.7

After Uhler and Bradley ( 1970 , Table 1)
Source: Agterberg ( 1974 , Table XVIII)

The latter probability is binomial. Multiplied by p this gives the negative binomial
distribution with:

p k 1 q k

r þ k 1

p k ¼

k

with generating function:

r

p

gs ðÞ ¼

1 qs

2 ( X )

rq / p 2 . For the negative binomial:

It follows that E ( X ) ¼ μ ¼ rq / p and ˃ ¼

2 ( X ) 2 2

˃ .

Comparison of the sample mean and variance of a set of discrete geological data
often provides a guideline as to which one of these two distribution (positive or
negative binomial) should be fitted ( cf . Ondrick and Griffiths 1969 ).
An example of a situation in which the negative binomial provides a better
approximation than the Poisson distribution is shown in Table 2.1 . It is based on a
study by Uhler and Bradley ( 1970 ) who divided the Alberta sedimentary basin into
8,811 cells, each measuring 5 miles (8 km) on a side., and counted the number of oil
deposits per cell. The results in Table 2.1 show that the negative binomial provides
the better fit. Obviously, the Poisson distribution that has only one parameter (

¼ ˃ >μ . On the other hand, for the ordinary (positive) binomial: ˃ <μ

ʻ )is

not sufficiently flexible in this application.
If r 1, the negative binomial distribution reduces to the geometric distribution

that can be illustrated as follows by an application to lithological components in the
Oficina Formation, eastern Venezuela. Krumbein and Dacey ( 1969 ) found that
the thicknesses of these components can be described by geometric distributions.
Their data consisted of a series of letters A, B, C, and D, originally obtained by
Scherer ( 1968 ) who coded the rock types as (A) sandstone, (B) shale, (C) siltstone,
and (D) lignite, in a well core at 2-ft (61 cm) intervals. Although the thicknesses of
these lithologies are continuous random variables, the sampling scheme reduced
them to discrete random variables.

¼