Digital Signal Processing Reference
In-Depth Information
P.2
The mean value of the mean value is the mean value itself.
Being
E
{
X
}
¼
const, from (
2.240
), it easily follows:
EfEfXgg ¼ EfXg:
(2.241)
P.3
The expected value of a random variable
X
multiplied by the constant a is
equal to the product of the constant a and the expected value of the normal
random variable:
E
{
aX
}
¼ aE
{
X
},
a ¼
const.
From (
2.230
), the expected value of
aX
, where
a
is a constant, is:
EfaXg¼
X
N
ax
i
PfX ¼ x
i
g¼a
X
N
x
i
PfX ¼ x
i
g¼aEfXg:
(2.242)
i¼
1
i¼
1
(Because
a
is constant it can be moved in front of the sum, which itself
represents
E
{
X
}).
P.4
This is a combination of P.1 and P.2:
E
{
aX
+
b
}
¼ aE
{
X
}+
b
Combining (
2.240
) and (
2.242
), we have:
EfaX þ bg¼
X
N
i¼
1
ðax
i
þ bÞPfX ¼ x
i
g
¼ a
X
x
i
PfX ¼ x
i
gþb
X
N
N
PfX ¼ x
i
g:
(2.243)
i¼
1
i¼
1
The first sum in (
2.243
)is
E
{
X
} while the second sum is equal to 1, yielding:
EfaX þ bg¼aEfXgþb:
(2.244)
P.5
The linear transformation
g
(
X
) of the random variable
X
and its expectation are
commutative operations.
From (
2.244
), it follows that if the function of the random variable
X
is a linear
function
gðXÞ¼aX þ b;
(2.245)
then the linear transformation of the random variable and its expectation are
commutative operations:
EgðXfg¼aEfXgþb ¼ gEfXg
ð
Þ:
(2.246)
However, in a general case in which the transformation
g
(
X
) is not linear
EfgðXÞg 6¼ gðEfXgÞ:
(2.247)
Search WWH ::
Custom Search