Encyclopedia of Environmental Science and Engineering, Volume I and II

1124 STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE

for the first, second, third, etc. moment, where f ( x i ) is the

probability function of the variable x. Moments need not be

taken around the mean of the distribution.

However, this is the most important practical case. The

first and second moments of a distribution are especially

important. The mean itself is the first moment and is the

most commonly used measure of central tendency for a dis-

tribution. The second moment about the mean is known as

the variance. Its positive square root, the standard deviation,

is a common measure of dispersion for most distributions.

For the binomial distribution the first moment is given by

μ = nu (4)

and the second moment is given by

suu

(^2)
n(). 1
(5)
The assumptions underlying the binomial distribution are that
the value of u is constant over trials, and that the trials are
independent; the outcome of one trial is not affected by the
outcome of another trial. Such trials are called Bernoulli trials.
The binomial distribution applies in the case of sampling with
replacement. Where sampling is without replacement, the
hypergeometric distribution is appropriate. A generalization
of the binomial, the multinomial, applies when more than two
outcomes are possible for a single trial.
The Poisson distribution can be regarded as the limit-
ing case of the binomial where n is very large and u is very
small, such that nu is constant. The Poisson distribution is
important in environmental work. Its probability function is
given by
fx
e
x
x
(; )
!
l ,
l l



(6)
where l = nu remains constant.
Its first and second moments are
m
 (7)
s^2
. (8)
The Poisson distribution describes events such as the
probability of cyclones in a given area for given periods of
time, or the distribution of traffic accidents for fixed periods
of time. In general, it is appropriate for infrequent events,
with a fixed but small probability of occurrence in a given
period. Discussions of discrete probability distributions can
be found in Freund among others. For a more extensive dis-
cussion, see Feller.
Continuous Distributions
The distributions mentioned in the previous section are all
discrete distributions; that is, they describe the distribution
of random variables which can be taken on only discrete
values.
Not all variables of interest take on discrete values; very
commonly, such variables are continuous. The analogous
function to the probability function of a discrete distribution
is the probability density function. The probability density
function for the standard normal distribution is given by
fx()
ex/.
1
2
(^22)
p
(9)
It is shown in Figure 3. Its first and second moments are
given by
m
p


 
1 
2
0
(^22)
∫ xex dx
(10)
and
s
p
2221
2
1
2


 

∫− xex d.x
(11)
(^05101520)
NUMBER OF X
F(X)
2.5
7.5
1.0
.5
FIGURE 2
–3 –2 –1 0 1 2 3
X(σ UNITS)
f(X)
0.0
0.1
0.2
0.3
0.4
FIGURE 3
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