Geoscience Reference
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behind each one of them. Hence the density function of the m th smallest observation
X m , occurring in conjunction with the occurrence of the remaining ( n
1) observations,
is equal to n times the probability given by (13.22), multiplied by the density of X m by
itself (Equation (13.21)), or
n !
1)! F m 1
F m ) n m
φ
( X m )
=
(1
f ( X m )
(13.23)
m
( n
m )!( m
It is known (see Mood and Graybill, 1963) that, if y is a function of z
,
y
=
y ( z ), and
if y has a density function f ( y ), then the density function of z is given by
f [ y ( z )]
d [ y ( z )]
dz
g ( z )
=
(13.24)
Applying this to the present case by putting y
=
X m and z
=
F m , one obtains the density
function of F m =
F m ( X m ) from (13.23), namely
n !
1)! F m 1
F m ) n m
g ( F m )
=
(1
(13.25)
m
( n
m )!( m
The mean of F m is the first moment, or, since 0
F m
1,
1
m 1 =
F m =
F m g ( F m ) dF m
0
or
1
n !
y m (1
y ) n m dy
F m =
(13.26)
( n
m )!( m
1)!
0
in which y is a dummy variable of integration. The integral in this expression is a complete
beta function, which can be expressed in terms of factorials (e.g. Abr amowitz and Stegun,
1964, p. 258) as m !( n
1), which
proves that the assumed plotting position of Equation (13.19) is in f act the mean of the
probability of non-exceedance of the m th observation, that is P m =
m )!
/
( n
+
1)!. This yields immediately F m =
m
/
( n
+
F m .
Probability graph paper
An empirical probability plot on graph paper with linear scales usually results in an
S-shaped curve, with considerable curvature. To facilitate interpretation of the plotted
data and interpolation, it is desirable to eliminate or reduce this curvature by stretching or
shrinking the scale in the appropriate range of values of P m , so that the points plot more
nearly along a straight line. A common way to accomplish this is the use of probability
paper. Hazen (1914b) appears to have been the first to advocate the use of probability
paper in hydrology.
Probability graph paper, for any given probability function F ( x ), is designed in such
a way that, when that function or the corresponding return period T r ( x ) is plotted against
x , one obtains a straight line. The most common types of probability paper involve a
probability function with two parameters, say a and b , whose values depend on the data
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