ANALYSIS OF WATER-QUALITY MEASUREMENTS - Water Quality Engineering in Natural Systems - page 339

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ANALYSIS OF WATER-QUALITY MEASUREMENTS

10.1 INTRODUCTION 10.2 PROBABILITY DISTRIBUTIONS

Water-quality data typically consist of measurements of
stochastic variables that can only be characterized by
probability distributions. This type of data is in contrast
to measurements of deterministic variables whose
values are invariable. Factors that cause water-quality
variables to be random include natural spatial and tem-
poral variability, as well as measurement anomalies.
Probability distributions describing stochastic vari-
ables constitute the bases for many of the methods used
to analyze water-quality data. However, the true prob-
ability distributions of water-quality variables are
seldom, if ever, known with certainty, and so they must
be estimated or approximated based on observed data.
The use of measured data to estimate the stochastic
properties of random variables is encompassed by the
field of statistics. As a consequence, an understanding of
the fundamentals of both probability and statistics is
generally required for appropriate application of the
data analysis methods commonly used in analyzing
water-quality data. Water-quality standards are some-
times stated in probabilistic terms, and in these cases,
probabilistic analysis of water-quality measurements
are indeed required.
The fields of probability and statistics are quite broad.
However, the conditions normally encountered in ana-
lyzing water-quality data are sufficiently narrow that a
more focused approach to the application of probability
and statistics to the analysis of water-quality data is
appropriate. It is the intent of the present chapter is to
provide such an exposition.

A probability function defines the relationship between
the outcome of a random process and the probability
of occurrence of that outcome. If the sample space, S ,
contains discrete elements, then the sample space is
called a discrete sample space ; if S is a continuum, the
sample space is called a continuous sample space . The
sample space of a random variable is commonly
denoted by an upper-case letter (e.g., X ), and the cor-
responding lower-case letter denotes an element of
the sample space (e.g., x ). Discrete probability distribu-
tions describe the probability of outcomes in discrete
sample spaces, while continuous probability distribu-
tions describe the probability of outcomes in continu-
ous sample spaces. Water-quality data are usually
drawn from a continuous sample space, and so the
emphasis of this chapter is on continuous probability
distributions.

10.2.1 Properties of Probability Distributions

If X is a random variable with a continuous sample
space, then there are an infinite number of possible
outcomes and the probability of occurrence of any
single value of X is zero. This problem is addressed by
defining the probability of an outcome being in the
range [ x , x + Δ x ] as f ( x )Δ x , where f ( x ) is the probability
density function . Based on this definition, any valid
probability density function must satisfy both of the
following conditions: