The expected value of a function
f·
f
·
of a random variable XX
is defined to be
EfX=∫-∞∞fxpXxdx
f
X
x
f
x
p
X
x
(1)
Several important quantities are expected values, with specific
forms for the function
f·
f
·
.
-
fX=X
f
X
X
. The expected value or mean of a
random variable is the center-of-mass of the probability
density function. We shall often denote the expected value
by m X
m X
or just
mm when the meaning is clear.
Note that the expected value can be a number never assumed
by the random variable (
pXm
p
X
m
can be zero). An important property of the
expected value of a random variable is
linearity:
EaX=aEX
a
X
a
X
, aa being a scalar.
-
fX=X2
f
X
X
2
.
EX2
X
2
is known as the mean squared value of
XX and represents the "power"
in the random variable.
-
fX=X-
m
X
2
f
X
X
m
X
2
. The so-called second central difference of a
random variable is its variance, usually
denoted by
σ
X
2
σ
X
2
. This expression for the variance simplifies to
σ
X
2
=EX2-EX2
σ
X
2
X
2
X
2
, which expresses the variance operator
σ·2
·
. The square root of the variance
σ X
σ X
is the standard
deviation and measures the spread of the distribution
of XX. Among all possible
second differences
X-c2
X
c
2
, the minimum value occurs when
c=
m
X
c
m
X
(simply evaluate the derivative with respect to
cc and equate it to
zero).
-
fX=Xn
f
X
X
n
.
EXn
X
n
is the
n th
n th
moment of the random variable and
EX-
m
X
n
X
m
X
n
the
n
th
n
th
central moment.
-
fX=ⅇⅈuX
f
X
u
X
. The characteristic function of a
random variable is essentially the Fourier Transform of the
probability density function.
EⅇⅈuX≡
Φ
X
ⅈu=∫-∞∞pXxⅇⅈuxdx
u
X
Φ
X
u
x
p
X
x
u
x
(2)
The moments of a random variable can be calculated from the
derivatives of the characteristic function evaluated at the
origin.
EXn=ⅈ-ndndun
Φ
X
ⅈu|u,u=0
X
n
n
u
0
u
n
n
Φ
X
u
(3)
