Characteristics of random variables

Mathematical expectation. Properties of mathematical expectation.

Variance. Properties of variance. Root-mean-square deviation.

Mathematical expectation of a discrete random variable  X , which receives a finite number of values  х_i  with probabilities  р_i , is called the sum:

М ( Х ) = х₁ · р₁ + х₂ · р₂ + х₃ · р₃ + ... + х_n · р_n .

Properties of mathematical expectation:

   1)   М ( с  · Х ) = с  · М ( Х ) ,   c R ,

   2)   М ( Х + Y ) = М ( Х ) + М ( Y ) ,     Х , Y Е ,

    3)   М ( Х  · Y ) = М ( Х ) · М ( Y )   for independent random variables  Х  and  Y . 

Variance of a random variable  X  is called the number:

 D ( Х ) = М{ [ Х – М ( Х )] ² }= М ( Х ² ) – [М ( Х )] ² .

Properties of variance:

   1)   D ( с  · Х ) = с ²  · D ( Х ) ,   c R , 

   2)   D ( Х + Y ) = D ( Х ) + D ( Y )   for independent random variables  Х  and  Y . 

Root-mean-square deviation:

Back