Titel_SS06

Standard Normal

Normal

X

Shift

Scaling

0

1

Y

Y









Standard Normal

Normal

X

Shift

Scaling

0

1

Y

Y









Figure 2.13: Illustration of the relationship between a Normal distributed random variable and a
standard Normal distributed random variable.

The Lognormal Distribution

A random variable Y is said to be lognormal distributed if the variable Zln( )Y is Normal
distributed. It thus follows that if an uncertain phenomenon can be assumed to originate from
a multiplicative effect of several uncertain contributions then the probability distribution for
the phenomenon can be assumed to be lognormal distributed.

The lognormal distribution has the property that if:

1

i

n a

i

i

PY



. (2.33)

and all are independent lognormal random variables with parameters Yi (i, )i and 'i 0 as
given in Table 2.7 then also is lognormal with parameters: P

1

n

P ii

i

((a



 (2.34)

22

1

n 2

P ii

i

) a)



 (2.35)

Properties of the Expectation Operator

It is useful to note that the expectation operation possess the following properties, where
and are constants and

ab,

c X is a random variable:

 

 

 

 12 () () 1 () 2 ()

Ec c

EcX cE X

Ea bX a bE X

Eg X g X Eg X Eg X







 

(2.36)

The implication of the last equation is that expectation, like differentiation or integration, is a
linear operation. This linearity property is useful since it can be used, for example, to find the
following formula for the variance of a random variable X in terms of more easily calculated
quantities: