Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 130

A.7.2.6 The Lognormal Distribution

lnX has a normal distribution.
The density of the distribution of X is then

X has a lognormal distribution ↔

x

p

Þ 1

2

f 1

2

f X x ðÞ ¼ ð 2 p exp½ ð ln ð 1 = 2 x =nÞÞ =f 1R þ x ðÞ;

for ʾ the median of the distribution of X and ʶ = Var(lnX).

2

EX ¼ n exp½ 1 =ð f ;

2

2

Var X ðÞ ¼ ðÞ EX ½exp ðf Þ 1 :

A.7.3 Examples of Multidimensional Distributions

For joint distributions, specially useful is to consider joint normal distributions.

A.7.3.1 Bidimensional Normal Distribution

The pair of random variables (X, Y) has a bidimensional normal distribution with
expected values

2
X

2
Y

ʼ X and ʼ Y , variances r and r and correlation coef cient ˁ XY ,

what is formally denoted by (X,Y) Normal( l X ; l Y ; r

2

X ; r

2

Y ; q XY ) if and only if

*

their joint density has the form

2 2

f XY x ðÞ ¼ ; y C 1 exp f C 2 ½½ ð x l X Þ=r X 2 q ½ ð x l X Þ=r X ½ ð y l Y Þ=r Y þ ½ ð y l Y Þ=r Y g

where

pr X r Y Þ 1

XY Þ 1 = 2

2

C 1 ¼ ð 2 ð 1 q

and

2

XY 1

C 2 ¼½ 2 ð 1 q :

So,

if q ¼ 0 ; X and Y are independent :

This extends to vectors of random variables.