Titel_SS06

Distribution type Parameters Moments

Beta, axb







 



11

1

() ()()

()

rt

X rt

rt x abx

fx

rt ba





+ 



++ 






11

1

() ()()

()

x rt

X rt

a

rt ua bu

Fx du

rt ba





+ 



++ 

0

0

a

b

r

t

&

&

a(ba)r

rt

 

1

ba rt

rt rt

 

Table 2.7: Probability distributions, Schneider (1994).

The Normal Distribution

The Normal probability distribution follows from the central limit theorem as a result of the
sum of independent (or almost) random variables. It is thus applied very frequently in
practical problems for the probabilistic modelling of uncertain phenomena which may be
considered to originate from a cumulative effect of several uncertain contributions.

The Normal distribution has the property that the linear combination S of Normal
distributed random variables

n

Xi,in1, 2,..., :

0

1

n

ii

i

Sa aX



  (2.31)

is also Normal distributed. The distribution is said to be closed in respect to summation.

One special version of the Normal distribution should be mentioned, namely the Standard
Normal distribution. In general a standardized (some times referred to as a reduced) random
variable is a random variable which has been transformed such that it has an expected value
equal to zero and a variance equal to one, i.e. the random variable Y defined by:

X

X

X

Y







 (2.32)

is a standardized random variable. If the random variable X follows and Normal distribution
the random variable Y is standard Normal distributed. In Figure 2.13 the process of
standardization is illustrated.

It is common practice to denote the cumulative distribution function for the standard Normal
distribution by ( ),x and the corresponding density function by ( )- x. These functions are
broadly available in software packages such as MS Excel and Matlab.