Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 56

Another discrete random variable to be used later in this topic (Chap. 9 ) has the
logarithmic series distribution with:

k

p k ¼ ʱˑ ð k ¼ 1, 2, ...; 0 < ˑ < 1 Þ:

k

e k )]/[log e /(1

Its moment generation function is: g ( s ) ¼ [log e /(1 ˑ ˑ )]. The

) 1 and ) 2 . first two moments about the origin are: E ( X ) ¼ ʱˑ (1 ˑ μ 2 ¼ ʱˑ (1 ˑ

2 ( X )

) 2 . This model Therefore, ˃ ¼ ʱˑ (1 ʱˑ )(1 ˑ is often used in the

biosciences for spatial distribution of species.

2.3 Continuous Frequency Distributions
and Statistical Inference

Continuous random variables can assume any value on the real line. Traditionally,
an important role is played by the normal or Gaussian distribution. It is frequently
observed in practice. Theoretically, it is the end product of the central-limit
theorem. The normal distribution underlies many methods of statistical inference
such as z -test, Student's t -test, chi-square test and analysis of variance.

Box 2.3: Moment Generating Function and Characteristic Function
If X is a continuous random variable, its moment generating function satisfies:
m ( u )

¼ R 1 1 ¼ R 1 1

E ( e uX ) e ux f ( x ) dx . Consequently,

0 r

x r f ( x ) dx m r (0); ¼ μ ¼

[ m 0 (0)] 2 . For continuous random variables,
characteristic functions g ( u ) have a wider field of application than moment
generating functions. They satisfy the following inverse relationship:

m 0 (0);

2 ( X )

m 00 (0) and E ( X ) ¼ ˃ ¼

Z 1 Z 1

1
2 ˀ

e iux fx e iux gx Ee iuX gu ðÞ ¼ ðÞ ¼ ðÞ dx ; fx ðÞ ¼ ðÞ dx . The moments

1 1

0
r

E ( X r ) i r g ( r ) (0). The Pareto distribution with frequency density

satisfy: μ ¼ ¼

ak a

fx provides an example. Provided that r is less

than a ,the r -th moment about zero is:

ðÞ ¼

x aþ 1 k

ð > 0, a > 0 ; x k Þ

0
r ¼

ak r
a

ak

μ

r . Consequently, EX

ðÞ ¼

1 and

a

a 2

2 ( X )

˃ ¼ . The Pareto will be used in Chaps. 3 and 11 .

2 a 2

ð a 1 Þ ð Þ

2.3.1 Central-Limit Theorem

The central-limit theorem evolved from the de Moivre-Laplace theorem which
stated that in the limit (sample size n

) a positive binomial distribution
becomes a normal (Gaussian) distribution ( cf . Bickel and Doksum 2001 , p. 470).

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