Random Variable - Random Signals and Processes Primer with MATLAB - page 90

From ( 2.277a ), using ( 2.72 ), we obtain:

Efbg¼b: (2.277b)

P.2 The mean value of the mean value is the mean value itself.
Being E { X } ¼ const, from ( 2.277b ), where b is switched with E { X }, we have:

EfEfXgg ¼ EfXg:

(2.278)

P.3 E { aX } ¼ aE { X }, a ¼ const.

From ( 2.273 ), the expected value of aX , where a is the constant, is:

1

axf X ðxÞ d x ¼a 1
1

EfaXg¼ xf X ðxÞ d x ¼aEfXg: (2.279)

1

(Since a is constant, it can be moved in front of the integral, which itself
represents E { X }).

P.4 E { aX + b } ¼ aE { X }+ b .

Combining ( 2.277b ) and ( 2.279 ), we have:

1

ðax þ bÞf X ðxÞ d x ¼ a 1
1

EfaX þ bg¼ xf X ðxÞ d x þ b

1

1

f X ðxÞ d x ¼ aEfXgþb: (2.280)

1

Note that the first integral on the right side of ( 2.280 ) represents the expected
value of the random variable X , while the second integral is equal to 1 (see the
characteristic of the density function ( 2.83 )). This results in:

EfaX þ bg¼aEfXgþb:

(2.281)

P.5 A linear function g ( X ) of the random variable X and its expectation are
commutative operations.

If the function of the random variable X is a linear function,

gðXÞ¼aX þ b; (2.282)

where a and b are constants, then from ( 2.281 ), we have:

EgðXf ¼ EfaX þ bg¼aEfXgþb ¼ gEfXg ð Þ: (2.283)