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Continuous variables: probability and
density function
Moving from discrete to continuous variables there are the
usual problems with infinite possibilities,
similar to those found in
Zeno's ``Achilles and the tortoise'' paradox.
In both cases
the answer is given by infinitesimal
calculus. But some comments are needed:
After this short introduction, here is a list of
definitions, properties and notations:
 Cumulative distribution function:

(4.26) 
or

(4.27) 
 Properties of and :

 Expectation value:

 Uniform distribution:
 ^{4.1}
:
Expectation value and standard deviation:
 Normal (Gaussian) distribution:

:

(4.34) 
where and (both real) are the expectation value and standard
deviation^{4.2},
respectively.
 Standard normal distribution:

the particular normal distribution of mean 0 and standard
deviation 1, usually indicated by :

(4.35) 
 Exponential distribution:

:
We use the symbol instead of because this distribution
will be applied to the time domain.
Survival probability:

(4.38) 
Expectation value and standard deviation:
The real parameter has the physical meaning of lifetime.
 Poisson
Exponential:

If (= ``number of counts during the time '') is
Poisson distributed then (= ``interval of time to wait 
starting from any instant  before the first count
is recorded'') is exponentially distributed:
Next: Distribution of several random
Up: Random variables
Previous: Discrete variables
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Giulio D'Agostini
20030515