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 õ i with probabilities ð i , is called the sum:
Ì ( Õ ) = õ 1 · ð 1 + õ 2 · ð 2 + õ 3 · ð 3 + ... + õ n · ð n .
Properties of mathematical expectation:
1)
Ì
(
ñ
·
Õ
) =
ñ
·
Ì
(
Õ
) ,
c
R
,
2)
Ì
(
Õ
+
Y
) =
Ì
(
Õ
) +
Ì
(
Y
) ,
Õ
,
Y
Å
,
3) Ì ( Õ · Y ) = Ì ( Õ ) · Ì ( Y ) for independent random variables Õ and Y .
Variance of a random variable X is called the number:
D ( Õ ) = Ì { [ Õ – Ì ( Õ )] 2 }= Ì ( Õ 2 ) – [ Ì ( Õ )] 2 .
Properties of variance:
1)
D
(
ñ
·
Õ
) =
ñ
2
·
D
(
Õ
) ,
c
R
,
2) D ( Õ + Y ) = D ( Õ ) + D ( Y ) for independent random variables Õ and Y .
Root-mean-square deviation:
