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## Discrete variables

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:
 (4.2) (4.3) (4.4)

Cumulative distribution function:

 (4.5)

Properties:
 (4.6) (4.7) (4.8) (4.9)

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:

 Var   E   E (4.13)

Standard deviation:

 (4.14)

Transformation properties:
 Var Var (4.15) (4.16)

Binomial distribution:

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

 (4.17)

Expectation value, standard deviation and variation coefficient:
 (4.18) (4.19) (4.20)

is often indicated by .
Poisson distribution:

:

 (4.21)

( is an integer, is real.)
Expectation value, standard deviation and variation coefficient:
 (4.22) (4.23) (4.24)

Binomial Poisson:

Next: Continuous variables: probability and Up: Random variables Previous: Random variables   Contents
Giulio D'Agostini 2003-05-15