Geoscience Reference
In-Depth Information
13
ELEMENTS OF FREQUENCY
ANALYSIS IN HYDROLOGY
One of the core questions in hydrologic data analysis is how to assign a probability
of future occurrence or a risk estimate to an event of a given magnitude, on the basis
of an available record of measurements. Over the years a number of concepts have
been developed for this purpose, which are not usually part of standard treatments of
elementary statistics. A few of these are reviewed in this chapter, together with some
other indispensable fundamentals.
13.1
RANDOM VARIABLES AND PROBABILITY
In practical terms, a random variable may be defined as the magnitude of an event, which
is the outcome of an experiment; the variable is called random because this magnitude
cannot be predicted with certainty. Random variables can be classified as discrete or as
continuous, or sometimes as a combination of the two.
When an event
A
occurs
n
A
times in an experiment, that is carried out
n
times, its
relative frequency is the ratio (
n
A
/
n
). The probability of this event can then be defined
as the limit of its relative frequency, when
n
is allowed to increase indefinitely, or
n
A
n
P
(
A
)
=
lim
(13.1)
n
→∞
To be sure, this definition is intuitively appealing, as it gives a “feel” for the meaning
of probability in everyday life. However, in any physical experiment the number
n
can
only be finite, so that this limit is only a hypothesis. For this reason, the probability
of an event is preferably defined axiomatically (Papoulis, 1965), as a number linked to
the event, with the properties, that (i) the probability cannot be negative, or
P
(
A
)
0;
(ii) the probability of all possible outcomes, i.e. the probability of certainty, equals unity;
(iii) if
A
and
B
are mutually exclusive events, the probability of
A
or
B
occurring equals
the sum of their probabilities, or
P
(
A
or
B
)
≥
P
(
B
). Clearly, the frequency
definition (13.1) satisfies these axioms. In this sense, the word
relative frequency
usually
refers to the empirical probability, that is the probability estimated from a finite sample,
which is presumably drawn from an infinitely large population.
A random variable is called discrete when it can assume only certain values
x
i
, with
=
P
(
A
)
+
i
...;then the probability, that a discrete random variable
X
will assume a given
value
x
i
, can be written as
=
1
,
2
,
p
i
=
P
{
X
=
x
i
}
(13.2)
