In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
E
[
f
(
X
)
]
=
E
[
f
(
μ
X
+
(
X
−
μ
X
)
)
]
≈
E
[
f
(
μ
X
)
+
f
′
(
μ
X
)
(
X
−
μ
X
)
+
1
2
f
″
(
μ
X
)
(
X
−
μ
X
)
2
]
.
Notice that
E
[
X
−
μ
X
]
=
0
, the 2nd term disappears. Also
E
[
(
X
−
μ
X
)
2
]
is
σ
X
2
. Therefore,
E
[
f
(
X
)
]
≈
f
(
μ
X
)
+
f
″
(
μ
X
)
2
σ
X
2
where
μ
X
and
σ
X
2
are the mean and variance of X respectively.
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,
E
[
X
Y
]
≈
E
[
X
]
E
[
Y
]
−
cov
[
X
,
Y
]
E
[
Y
]
2
+
E
[
X
]
E
[
Y
]
3
var
[
Y
]
Analogously,
var
[
f
(
X
)
]
≈
(
f
′
(
E
[
X
]
)
)
2
var
[
X
]
=
(
f
′
(
μ
X
)
)
2
σ
X
2
.
The above is using a first order approximation unlike for the method used in estimating the first moment. It will be a poor approximation in cases where
f
(
X
)
is highly non-linear. This is a special case of the delta method. For example,
var
[
X
Y
]
≈
var
[
X
]
E
[
Y
]
2
−
2
E
[
X
]
E
[
Y
]
3
cov
[
X
,
Y
]
+
E
[
X
]
2
E
[
Y
]
4
var
[
Y
]
.
