Continuous random variable

A discrete random variable X has a finite number of possible values.

Random variable X is called discrete if there is such non-negative function

(1)

which puts in conformity to value х_i of variable X probability р_i with which it accepts this value.

Random variable X is called continuous if for any a < b there is such non-negative function

f (x), that (2)

Function f (x) is called density function of continuous random variable.

Probability of that random variable X (discrete or continuous) accepts value, smaller х, is called distribution function of random variable X and is designated F (x):

(3)

Distribution function is universal kind of distribution law, suitable for any random variable.

Basic properties of distribution function:

1) - not decreasing function, i.e. at ;

2) - limited function: ;

3) ;

4) ;

5) - continuous on the left function.

Except for this universal, there are also private kinds of distribution laws: series of distribution (only for discrete random variables) and density function (only for continuous random variables).

Basic properties of density function:

1) ;

2) . (4) Random variables that can assume values corresponding to any of the points contained in one or more intervals are called continuous.

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