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Random variables


Random variable. Discrete and continuous random variables.

Independent random variables. Density function.

Distribution function. General properties of distribution function.


A variable is called  random, if it can receive real values with definite probabilities as a result of experiment. The random variable  X  is called  discrete, if such non-negative function exists



which determines the correspondence between a value  i  of the variable   and the probability  i , that  X  receives this value.

Discrete random variables  X  and  Y  are called  independent random variables, if the events  = i  and  Y = yj  are independent for arbitrary  i  and  j .

The random variable X is called a continuous random variable, if for any numbers  a < b  such non-negative function  f ( x ) exists, that


The function   f ( x )  is called a density function of continuous random variable. The probability of the fact that a random variable  X  receives a value less than  x , is called a distribution function of random variable  X  and marked as  F ( x ) :


F ( x ) = (  X    x ) .


General properties of distribution function:


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