62
Statistical Methods for Estimating Petroleum Resources
lognormal, Pareto, and Weibull superpopulations. For the lognormal
case, a population with µ = 0 and σ
2 = 5 was assumed. The truncated
and shifted Pareto population with shape factor θ = 0.4, maximum
pool size = 4000, and minimum pool size = 1 was created. The Weibull
population with λ = 20, θ = 1.0 was generated for the current study. The
fi rst mixed population was created by mixing two lognormal populations. Parameters for population I are µ = 0, σ
2 = 3, and N 1 = 150. For
population II, µ = 3.0, σ
2 = 3.2, and N 2 = 150.
The second mixed population was generated by mixing lognormal
(N 1 = 100), Pareto (N 2 = 100), and Weibull (N 3 = 100) populations with a
total of 300 pools. In addition, a gamma distribution was also used for
reference.
The lognormal distribution is J-shaped if an arithmetic scale is used
for the horizontal axis, but it shows an almost symmetrical pattern
when a logarithmic scale is applied. The probability density function of
a lognormal distribution is defi ned as
2
2
1
l n
)
1
( )
exp 2
2
x
f x
x
−
=
−
m
s
s
p
(4.1)
where x is the pool size, µ is the mean of the logarithmic transformed
data, and σ
2 is the variance of the logarithmic transformed data.
The Weibull population displays a J-shaped distribution if the data
are plotted on an arithmetic scale, whereas it is almost symmetric but
skewed toward the left when plotted on a logarithmic scale. The probability density function of a Weibull distribution is defi ned as
−
=
−
( 1)
( )
exp
x
f x
x
a
a
a
a
b
b
(4.2)
where x is the pool size, with α (shape factor) > 0, and β (spread
factor) > 0.
The histograms of gamma and Pareto distributions display J-shaped
distributions on both arithmetic and logarithmic scales. The probability density function of a gamma distribution is defi ned as
1 exp
( )
( )
x
x
f x
G
−
−
=
a
a
b
b a
(4.3)
Statistical Methods for Estimating Petroleum Resources
lognormal, Pareto, and Weibull superpopulations. For the lognormal
case, a population with µ = 0 and σ
2 = 5 was assumed. The truncated
and shifted Pareto population with shape factor θ = 0.4, maximum
pool size = 4000, and minimum pool size = 1 was created. The Weibull
population with λ = 20, θ = 1.0 was generated for the current study. The
fi rst mixed population was created by mixing two lognormal populations. Parameters for population I are µ = 0, σ
2 = 3, and N 1 = 150. For
population II, µ = 3.0, σ
2 = 3.2, and N 2 = 150.
The second mixed population was generated by mixing lognormal
(N 1 = 100), Pareto (N 2 = 100), and Weibull (N 3 = 100) populations with a
total of 300 pools. In addition, a gamma distribution was also used for
reference.
The lognormal distribution is J-shaped if an arithmetic scale is used
for the horizontal axis, but it shows an almost symmetrical pattern
when a logarithmic scale is applied. The probability density function of
a lognormal distribution is defi ned as
2
2
1
l n
)
1
( )
exp 2
2
x
f x
x
−
=
−
m
s
s
p
(4.1)
where x is the pool size, µ is the mean of the logarithmic transformed
data, and σ
2 is the variance of the logarithmic transformed data.
The Weibull population displays a J-shaped distribution if the data
are plotted on an arithmetic scale, whereas it is almost symmetric but
skewed toward the left when plotted on a logarithmic scale. The probability density function of a Weibull distribution is defi ned as
−
=
−
( 1)
( )
exp
x
f x
x
a
a
a
a
b
b
(4.2)
where x is the pool size, with α (shape factor) > 0, and β (spread
factor) > 0.
The histograms of gamma and Pareto distributions display J-shaped
distributions on both arithmetic and logarithmic scales. The probability density function of a gamma distribution is defi ned as
1 exp
( )
( )
x
x
f x
G
−
−
=
a
a
b
b a
(4.3)