Geoscience Reference
In-Depth Information
In its later, more general form the theorem states roughly that the sum of many
independent random variables will be approximately normally distributed if
each summand has high probability of being small (Billingsley
1986
, p. 399).
An example of how averaging of data with different kinds of frequency distribu-
tions is shown in Fig.
2.1
. Because of the central-limit theorem, the averages of
n
measurements produce new random variables that become normally distributed
when
n
increases. Because in Nature and also in the social sciences so many random
variables are approximately normally distributed, and the central-limit theorem
seems to provide a plausible explanation, the normal distribution became a corner
stone of mathematical statistics.
Box 2.4: The Normal or Gaussian Distribution
If
X
is a normally distributed random variable with expected value equal
to
2
,
μ
and
variance
˃
its
frequency
density
f
(
x
)
satisfies:
n
2
˃
x
μ
˃
1
1
2
fx
ðÞ
¼
exp
g
and its cumulative frequency distribution is:
p
2
ˀ
Z
x
n
2
˃
x
μ
˃
1
1
2
Fx
ðÞ
¼
exp
g
dx
. The corresponding equations for
the
p
2
ˀ
1
random variable
Z
¼
(
X
μ
)/
˃
, which is the normal distribution in standard form, are:
Z
z
2
z
2
and
1
2
z
2
1
2
e
1
2
e
ˆ
ðÞ
¼
z
ʦ
ðÞ
¼
z
dz
.
p
p
ˀ
ˀ
1
Because of symmetry of the Gaussian density function with respect to
z
¼
0:
ʦ
(
z
)
¼
1
ʦ
(
z
). The graph of
ʦ
(
z
) is S-shaped. The fractiles
Z
P
of the standard
normal random variable satisfy
P
. Tables of these fractiles are widely
available.
P
is called the probit of Z
P
. It is noted that, originally, the term “probit”
was coined by Bliss (
1935
) for a fractile augmented by 5 in order to avoid the use of
negative numbers. To-day, however, the term “probit transformation” is widely
observed frequencies into their corresponding
Z
-values.
The sum of a number (
f
)of
Z
2
values is distributed as
ˇ
ʦ
(
Z
P
)
¼
2
(
f
) representing the
chi-square distribution for
f
degrees of freedom, which is a particular form of
the gamma distribution with probability density:
h
i
Þ
ʱ
1
exp
x
ʳ
ʲ
ð
x
ʳ
fx
ðÞ
¼
ð
ʱ >
0,
ʲ >
0,
x
> ʳ
Þ
ʲ
ʱ
ʓðÞ
Later in this topic, the gamma distribution will be used to model biostratigraphic
events (Sect.
9.3
) and amounts of rock types contained in grid cells superimposed
on geological maps (Sect.
12.8
). The main use of the gamma distribution, however,
is that it becomes
2
(
f
)if
ˇ
ʱ
¼
f
/2,
ʲ
¼
2 and
ʳ
¼
0.
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