Maximum Likelihood, Lognormality and Compound Distributions - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 99

The generating function g ( X i ) for Merriespruit is according to a straight line
whose prolongation would intersect the X -axis at a point with negative abscissa
ʱ 55. The second example (Muskox sulphides) gives a function g ( X i ) that resem-

bles a broken line. It would suggest a rather abrupt change in the generating process
for copper concentration after the value X i 0.5 %. The in￿uence of X i on dX i may

have decreased with increasing X i .
The usefulness of a graphical method of this type is limited, in particular because
random ￿uctuations in the original frequency histograms cannot be considered
carefully. However, the method is rapid and offers suggestions with respect to the
physico-chemical processes that may underlie a frequency distribution.

3.3 Compound Random Variables

Applications of the theory of compound random variables are helpful in situations
that geological entities occurring at points have properties such as size or volume
that can be modeled as continuous random variables. This topic is useful in mineral
potential evaluation where the targets of interest are ore deposits or oil pools that
are randomly distributed in a study area according to a Poisson or negative binomial
distribution but have sizes that satisfy, for example, a lognormal or Pareto
distribution.

3.3.1 Compound Frequency Distributions and Their
Moments

Suppose that K random variables X i have the same frequency distribution and
generating function g x ( s ). Their sum Y is a random variable with K terms. If
K and the X i are discrete random variables, the probability that Y is equal to the
integer number j is equal to the sum of the probabilities p 1 · P ( X 1 ¼ j ), p 2 · P ( X 1 ¼ j ),

where the p 1 , p 2 , ... , represent the probabilities that K is equal to 1, 2, ... , or:

X i ¼0 p i

PY ð ¼ j Þ ¼ PX 1 þ ð X 2 þþ X i ¼ j Þ

j ) are both equal to
zero. If g y ( s ) and g k ( s ) are the generating functions of Y and K , then:

The summation may start at i ¼ 0, for which P ( X 0 ) and P ( Y ¼

n i

1

g y ðÞ ¼ s p i g xðÞ ¼ g k f g x ðÞ s g

i ¼0