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Uncertain numbers are numbers in respect of which
we are in a condition of uncertainty. They can be the
number associated with the outcome of a die, to the number
which will be read on a scale when a measurement
is performed, or to the numerical
value of a physics quantity. In the sequel, we will
call uncertain numbers also ``random variables'',
to come close to what physicists are used to, but one should
not think, then, that ``random variables'' are only associated
with the outcomes of repeated experiments.
Stated simply, to define a random variable
means to find a rule which allows a real number
to be related univocally
(but not necessarily biunivocal)
to an event ().
One could write this expression
.
Discrete variables assume a countable range, finite or not.
We shall indicate
the variable
with
and
its numerical realization
with ;
and differently from
other notations, the symbol
(in place of or ) is also used for discrete variables.
Here is a list of definitions, properties and notations:
 Probability function:

To each possible value of we associate a degree of belief:

(4.1) 
, being a probability, must satisfy the following properties:
 Cumulative distribution function:


(4.5) 
Properties:
 Expectation value (mean):

E 
(4.10) 
In general, given a function of ,
E 
(4.11) 
E is a linear operator:
E E 
(4.12) 
 Variance and standard deviation:

Variance:
Standard deviation:

(4.14) 
Transformation properties:
 Binomial distribution:

(hereafter ``'' stands for ``follows'');
stands for binomial with parameters
(integer) and (real):

(4.17) 
Expectation value, standard deviation and variation coefficient:
is often indicated by .
 Poisson distribution:

:

(4.21) 
( is an integer, is real.)
Expectation value, standard deviation and variation coefficient:
 Binomial
Poisson:

Next: Continuous variables: probability and
Up: Random variables
Previous: Random variables
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Giulio D'Agostini
20030515