Geoscience Reference
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(2.8)
A frequentist who has no data is paralysed (Cullen and Frey, 1999). However, there may
be cases in which data are lacking in quantity or quality but for which an analyst has
other information that can be used to construct a probabilistic representation of an input
to a model. This is the kind of a situation the Baysian approach provides advantage over
the frequentist approach. But this approach is not without shortcomings. In using a
conventional Bayesian analysis the information concerning uncertain statistical
parameters or the states of nature, no matter how vague, must necessarily be represented
by conventional, exactly specified, probability distributions. Caselton and Luo, (1992)
argues that such statistical precision may sometimes lead to the danger that
inappropriately strong conclusions being drawn from the decision analysis.
Probability density function
In probability theory the uncertainty is represented by a Probability Density Function
(PDF),
p
X
(x),
where
X
is the uncertain variable and
x
is its value. A cumulative form of
the PDF is called a Cumulative Distribution Function (CDF),
P
X
(x)
(Fig. 2.3). The
probability of
X
in the interval
(a, b]
is given by
(2.9)
It then follows
(2.10)
It also follows, if
P
X
(x)
has a first derivative, that
(2.11)
It should be noted that any function representing a probability distribution of a random
variable must necessarily satisfy the axioms of probability (Equations (2.1)—(2.3)).
Therefore, a CDF possesses following properties (Ang and Tang, 1975):
1.
P
X
(!")=0;
P
X
(+")=1.0
2.
P
X
(x)
!0, and is nondecreasing with
x,
and
3. It is continuous with
x
.
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