Theorem The exponentiation of a N (µ, σ 2 ) random variable is a log normal(α, β) random
variable.
Proof Let the random variable X have the normal distribution with probability density
function
1 x−µ 2
1
− ∞ < x < ∞.
fX (x) = √
e− 2 ( σ )
2πσ
The transformation Y = g(X) = eX is a 1–1 transformation from X = {x | − ∞ < x < ∞}
to Y = {y | y > 0} with inverse X = g −1 (Y ) = ln(Y ) and Jacobian
dX
1
= .
dY
Y
Therefore by the transformation technique, the probability density function of Y is
fY (y) = fX g −1 (y)
=
√
1
2πσ
!
dx
dy
e− 2 (
1
) 1
y
ln(y)−µ 2
σ
y > 0.
Let µ = ln(α) and σ = β. Then
fY (y) =
1
√
2πyβ
!
e− 2 (
1
ln(y/α) 2
β
)
y > 0,
which is the probability density function of the log normal distribution.
APPL Verification: The APPL statements
X
g
Y
Z
:=
:=
:=
:=
NormalRV(mu, sigma);
[[x -> exp(x)],[-infinity, infinity]];
Transform(X, g);
LogNormalRV(mu, sigma);
yield identical probability density functions for Y and Z.
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