Basic Notions of Probability Theory - Probabilistic Composition of Preferences, Theory and Applications - page 129

A.7.2.3 Exponential Distribution

e a x 1 R þ x X Exponential ðaÞ$ f X x ðÞ ¼ a ðÞ:

For X with an exponential distribution,

¼ a 1 EX :

A.7.2.4 Pareto Distribution

ðÞ ¼½ ba b =ð x bþ 1

X Pareto ða; bÞ$ f X x Þ 1 ½ a;þ/Þ x ðÞ for a [ 0e b [ 0 :

For X with a Pareto distribution,

EX ðÞ ¼ ab=ðb 1 Þ; for b [ 1 ;

and

E ðÞ ¼ þ/; for b 1 :

X Pareto ða; b; Þ$ ln ð X =aÞ Exponential ðbÞ:

A.7.2.5 Normal Distribution

X has a normal distribution with location parameter ʼ and dispersion parameter ˃

ðÞ ¼ ð

p

Þ 1

2

2

r 1

2

ð X Normal ðl; r ÞÞ $ f X x 2 p exp½ ðð 1 = 2 x lÞ =r :

EX ¼ l:

2

Var X ðÞ ¼ r :

The normal distribution has the following useful properties:

2

X Normal ðl; r Þ$ð X lÞ=r Normal 0 ð ; ; 1

any linear combination of random variables with a normal distribution has a normal
distribution;
if X n is the sum of n random variables independent and identically distributed
with a distribution with expected value ʼ and standard deviation ˃ then the

p Þ

distribution of .

As a particular instance of this last property, for the summands with Bernoulli
distributions, we have

ð X n n lÞ=ðr approaches a Normal(0,1) distribution as n !∞

p

nq 1 X Binomial n ð ; q Þ! ð X nq Þ= ½ ð q Þ Normal 0 ðÞ ; 1