Random Variable - Random Signals and Processes Primer with MATLAB - page 103

P.2 If a random variable is multiplied by any constant a , then its variance is
multiplied by the squared value of a : VAR faXg¼a 2 VAR fXg .

From the definition of the variance ( 2.279 ), we have:

n o

:

2

VAR ðaXÞ¼E

ðaX EfaXgÞ

(2.353)

Keeping in mind that EfaXg¼aEfXg (see ( 2.279 )), from ( 2.353 ) it follows:

n o

;

2

VAR ðaXÞ¼E ðaX aEfXgÞ

n o ¼ a 2 E n o ¼ a 2 VAR fXg:

2 2

¼ Ea 2

ðX EfXgÞ ðX EfXgÞ

(2.354)

P.3 This is a combination of P.1 and P.2: VAR faX bg¼ VAR faXg¼
a 2 VAR fXg:

From ( 2.352 ) and ( 2.353 ), the variance of the linear transformation of the r.v. X ,
where a and b are constants, is:

VAR faX þ bg¼ VAR faXg¼a 2 VAR fXg:

(2.355)

From ( 2.355 ), we note that

VAR faX þ bg 6¼ a VAR fXgþb: (2.356)

Therefore, as opposed to the mean value, in which the mean of linear transfor-
mation of the random variable results in the same linear transformation of its mean,
the relation ( 2.356 ) indicates that the variance of a linearly transformed r. v. X does
not result in a linear transformation of the variance of X . The reason for this is
because the variance is not a linear function, as evident from ( 2.353 ).

2.8.2.3 Standarized Random Variable

Properties ( 2.352 ) and ( 2.354 ) can be used to normalize the variable (i.e., to obtain
the corresponding random variable with zero mean and the variance of one).
Such random variable is called the standardized random variable . The use of

standardized random variables frequently simplifies discussion. The

standardization of random variables eliminates the effects of origin and scale.
Consider a random variable X with mean value m X and variance s 2 ; then the
random variable

Y ¼ X m X (2.357)