Page 54 - IJPS-8-2

International Journal of
Population Studies Used versus Offered densities of human population

Appendix Appendix C: Log-normal distributions and their
basic properties
Appendix A: Nomenclature
The log-normal distribution
z zone, a spatial entity in set Z covering the territory under
study The log-normal distribution is especially well-suited
to consumption models of two kinds: Power laws, on the
A , ground area of zone z first hand (e.g., Cowell, 2009), and log-normal CDFs, on
z
P , population in zone z the other hand. The latter kind has been used by Cramer
z
A , overall ground area of territory (1962) to study the diffusion of car motorization among
z
a population of households. Here, we shall focus on the
P , overall population in territory former kind, with some power r that needs not be an
z
a , unit ground area integer:
1
,
o a land unit cx c . x r (C-1)
1
P , population in o Of course, factor c needs be nonnegative to make some
o
1
O, total number of land units (equal to A / a ) sense.
1
Z
U, total number of people in territory (equal to P ,) Basic properties of log-normal distributions
Z
x, density level Let us recall the definition of a unidimensional log-
normal distribution: A real random variable X is said to
f & F , PDF & CDF of x regarding land units, with mean be distributed LN(, )ms 2 if it is positive and its natural
O
O
x and relative dispersion γ O logarithm is Gaussian, that is, ln()X ∼ N(, )m s . Denoting
2
O
f & F , PDF & CDF of x regarding people, with mean x as Φ the CDF of a reduced Gaussian variable and
U
U
U
and relative dispersion γ U t exp t 2 /2 / 2 the associated PDF, and letting
L, Lorenz function t (ln x ms)/ , the following outcomes are derived
x
successively in a straightforward way (e.g., Cowell,
G, Gini index 2009):
Appendix B: Consumption model F x ()
t
A consumption model can be stated in a generic fashion as O 1 x 1
.

follows. It relies on a consumption function say F exp(ms )
O
cx:  c x ( ), which takes nonnegative real values and 1
x
measures the amount of consumption made by an f sx. t ()
O
x
individual with attribute x. 1
2
O
Let then f denote the PDF of attribute x in the statistical E x exp m ( 2 s )
O
population of such individuals. The consumed units of all 2
O
2
the individuals make up a statistical population of their V x E x ( exp s 1( ) )
O
own, with PDF function f that satisfies the following
U
relation: exp()s 2 1
f c x().( (B-1) O 2
f x)
x
O
U

Postulating that the consumption function is Hence s ln(1 O ) .
r
monotonous, then eqn. (B-1) can be demonstrated using Furthermore, any derived random variable Y ≡ c . X
1
0
the same proof as for Equation (13). The proportionality with c > is a log-normal variable, too. This is because
1
coefficient is the reciprocal of c xdx . Thus Y ≥ 0 and ln Y ln c 1 r .( )Xln , implying that
O
f
cx
O
c
1 ln()Y N(ln r . ,( ))mrs 2 , making Y an LN variable
1
f c x().( (B-2)
x
f x)
.
2
U
c O O with parameters ln c rm and ()rs .
1
Volume 8 Issue 2 (2022) 48 https://doi.org/10.36922/ijps.v8i2.297