First moment
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Given
μ
X
{\displaystyle \mu _{X}}
σ
X
2
{\displaystyle \sigma _{X}^{2}}
X
{\displaystyle X}
[1] expected value of
f
(
X
)
{\displaystyle f(X)}
E
[
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=
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=
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E
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E
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2
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.
{\displaystyle {\begin{aligned}\operatorname {E} \left[f(X)\right]&{}=\operatorname {E} \left[f\left(\mu _{X}+\left(X-\mu _{X}\right)\right)\right]\\&{}\approx \operatorname {E} \left[f(\mu _{X})+f'(\mu _{X})\left(X-\mu _{X}\right)+{\frac {1}{2}}f''(\mu _{X})\left(X-\mu _{X}\right)^{2}\right]\\&{}=f(\mu _{X})+f'(\mu _{X})\operatorname {E} \left[X-\mu _{X}\right]+{\frac {1}{2}}f''(\mu _{X})\operatorname {E} \left[\left(X-\mu _{X}\right)^{2}\right].\end{aligned}}}
Since
E
[
X
−
μ
X
]
=
0
,
{\displaystyle E[X-\mu _{X}]=0,}
E
[
(
X
−
μ
X
)
2
]
{\displaystyle E[(X-\mu _{X})^{2}]}
σ
X
2
{\displaystyle \sigma _{X}^{2}}
E
[
f
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X
)
]
≈
f
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μ
X
)
+
f
″
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μ
X
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2
σ
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{\displaystyle \operatorname {E} \left[f(X)\right]\approx f(\mu _{X})+{\frac {f''(\mu _{X})}{2}}\sigma _{X}^{2}}
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions . For example,
E
[
X
Y
]
≈
E
[
X
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E
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Y
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−
cov
[
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Y
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E
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2
+
E
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E
[
Y
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3
var
[
Y
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{\displaystyle \operatorname {E} \left[{\frac {X}{Y}}\right]\approx {\frac {\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]}}-{\frac {\operatorname {cov} \left[X,Y\right]}{\operatorname {E} \left[Y\right]^{2}}}+{\frac {\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]^{3}}}\operatorname {var} \left[Y\right]}
Second moment
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Similarly,[1]
var
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{\displaystyle \operatorname {var} \left[f(X)\right]\approx \left(f'(\operatorname {E} \left[X\right])\right)^{2}\operatorname {var} \left[X\right]=\left(f'(\mu _{X})\right)^{2}\sigma _{X}^{2}-{\frac {1}{4}}\left(f''(\mu _{X})\right)^{2}\sigma _{X}^{4}}
The above is obtained using a second order approximation, following the method used in estimating the first moment. It will be a poor approximation in cases where
f
(
X
)
{\displaystyle f(X)}
delta method .
Indeed, we take
E
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)
]
≈
f
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μ
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+
f
″
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μ
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)
2
σ
X
2
{\displaystyle \operatorname {E} \left[f(X)\right]\approx f(\mu _{X})+{\frac {f''(\mu _{X})}{2}}\sigma _{X}^{2}}
With
f
(
X
)
=
g
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X
)
2
{\displaystyle f(X)=g(X)^{2}}
E
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Y
2
]
{\displaystyle \operatorname {E} \left[Y^{2}\right]}
var
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Y
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=
E
[
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−
μ
Y
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{\displaystyle \operatorname {var} \left[Y\right]=\operatorname {E} \left[Y^{2}\right]-\mu _{Y}^{2}}
An example is,
var
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≈
var
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E
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E
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cov
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+
E
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E
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var
[
Y
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.
{\displaystyle \operatorname {var} \left[{\frac {X}{Y}}\right]\approx {\frac {\operatorname {var} \left[X\right]}{\operatorname {E} \left[Y\right]^{2}}}-{\frac {2\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]^{3}}}\operatorname {cov} \left[X,Y\right]+{\frac {\operatorname {E} \left[X\right]^{2}}{\operatorname {E} \left[Y\right]^{4}}}\operatorname {var} \left[Y\right].}
The second order approximation, when X follows a normal distribution, is:[2]
var
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{\displaystyle \operatorname {var} \left[f(X)\right]\approx \left(f'(\operatorname {E} \left[X\right])\right)^{2}\operatorname {var} \left[X\right]+{\frac {\left(f''(\operatorname {E} \left[X\right])\right)^{2}}{2}}\left(\operatorname {var} \left[X\right]\right)^{2}=\left(f'(\mu _{X})\right)^{2}\sigma _{X}^{2}+{\frac {1}{2}}\left(f''(\mu _{X})\right)^{2}\sigma _{X}^{4}+\left(f'(\mu _{X})\right)\left(f'''(\mu _{X})\right)\sigma _{X}^{4}}
First product moment
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To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows. First, note that
cov
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,
f
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=
E
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f
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]
−
E
[
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]
E
[
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]
{\displaystyle \operatorname {cov} \left[f(X),f(Y)\right]=\operatorname {E} \left[f(X)f(Y)\right]-\operatorname {E} \left[f(X)\right]\operatorname {E} \left[f(Y)\right]}
E
[
f
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)
]
{\displaystyle \operatorname {E} \left[f(X)\right]}
E
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f
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)
f
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]
{\displaystyle \operatorname {E} \left[f(X)f(Y)\right]}
f
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f
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{\displaystyle f(X)f(Y)}
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{\displaystyle {\begin{aligned}f(X)f(Y)&{}\approx f(\mu _{X})f(\mu _{Y})+(X-\mu _{X})f'(\mu _{X})f(\mu _{Y})+(Y-\mu _{Y})f(\mu _{X})f'(\mu _{Y})+{\frac {1}{2}}\left[(X-\mu _{X})^{2}f''(\mu _{X})f(\mu _{Y})+2(X-\mu _{X})(Y-\mu _{Y})f'(\mu _{X})f'(\mu _{Y})+(Y-\mu _{Y})^{2}f(\mu _{X})f''(\mu _{Y})\right]\end{aligned}}}
Taking expectation of the above and simplifying—making use of the identities
E
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X
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)
=
var
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X
)
+
[
E
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]
2
{\displaystyle \operatorname {E} (X^{2})=\operatorname {var} (X)+\left[\operatorname {E} (X)\right]^{2}}
E
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X
Y
)
=
cov
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X
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Y
)
+
[
E
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)
]
[
E
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]
{\displaystyle \operatorname {E} (XY)=\operatorname {cov} (X,Y)+\left[\operatorname {E} (X)\right]\left[\operatorname {E} (Y)\right]}
E
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f
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f
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]
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{\displaystyle \operatorname {E} \left[f(X)f(Y)\right]\approx f(\mu _{X})f(\mu _{Y})+f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)+{\frac {1}{2}}f''(\mu _{X})f(\mu _{Y})\operatorname {var} (X)+{\frac {1}{2}}f(\mu _{X})f''(\mu _{Y})\operatorname {var} (Y)}
cov
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=
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{\displaystyle {\begin{aligned}\operatorname {cov} \left[f(X),f(Y)\right]&{}\approx f(\mu _{X})f(\mu _{Y})+f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)+{\frac {1}{2}}f''(\mu _{X})f(\mu _{Y})\operatorname {var} (X)+{\frac {1}{2}}f(\mu _{X})f''(\mu _{Y})\operatorname {var} (Y)-\left[f(\mu _{X})+{\frac {1}{2}}f''(\mu _{X})\operatorname {var} (X)\right]\left[f(\mu _{Y})+{\frac {1}{2}}f''(\mu _{Y})\operatorname {var} (Y)\right]\\&{}=f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)-{\frac {1}{4}}f''(\mu _{X})f''(\mu _{Y})\operatorname {var} (X)\operatorname {var} (Y)\end{aligned}}}
Random vectors
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See also
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^ a b Haym Benaroya, Seon Mi Han, and Mark Nagurka. Probability Models in Engineering and Science . CRC Press, 2005, p166.
^ Hendeby, Gustaf; Gustafsson, Fredrik. "ON NONLINEAR TRANSFORMATIONS OF GAUSSIAN DISTRIBUTIONS" (PDF) . Retrieved 5 October 2017 . ^ Rego, Bruno V.; Weiss, Dar; Bersi, Matthew R.; Humphrey, Jay D. (14 December 2021). "Uncertainty quantification in subject‐specific estimation of local vessel mechanical properties" . International Journal for Numerical Methods in Biomedical Engineering . 37 (12): e3535. doi :10.1002/cnm.3535 . ISSN 2040-7939 . PMC 9019846 PMID 34605615 .
Further reading
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![{\displaystyle {\begin{aligned}\operatorname {E} \left[f(X)\right]&{}=\operatorname {E} \left[f\left(\mu _{X}+\left(X-\mu _{X}\right)\right)\right]\\&{}\approx \operatorname {E} \left[f(\mu _{X})+f'(\mu _{X})\left(X-\mu _{X}\right)+{\frac {1}{2}}f''(\mu _{X})\left(X-\mu _{X}\right)^{2}\right]\\&{}=f(\mu _{X})+f'(\mu _{X})\operatorname {E} \left[X-\mu _{X}\right]+{\frac {1}{2}}f''(\mu _{X})\operatorname {E} \left[\left(X-\mu _{X}\right)^{2}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9453f100161227140288817a070c40ad87c22b2)
![{\displaystyle E[X-\mu _{X}]=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/000acb6340500505b2c39bf53b182169aa6b85d7)
![{\displaystyle E[(X-\mu _{X})^{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1906e7bfddbbb4fa3b71cf84887c95cb180995)
![{\displaystyle \operatorname {E} \left[f(X)\right]\approx f(\mu _{X})+{\frac {f''(\mu _{X})}{2}}\sigma _{X}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27108a7bd1d981b02fa89424930f479ec6fe8d4e)
![{\displaystyle \operatorname {E} \left[{\frac {X}{Y}}\right]\approx {\frac {\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]}}-{\frac {\operatorname {cov} \left[X,Y\right]}{\operatorname {E} \left[Y\right]^{2}}}+{\frac {\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]^{3}}}\operatorname {var} \left[Y\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8b82f9c15c42b3fa41c397dd3b6a1d67735539)
![{\displaystyle \operatorname {var} \left[f(X)\right]\approx \left(f'(\operatorname {E} \left[X\right])\right)^{2}\operatorname {var} \left[X\right]=\left(f'(\mu _{X})\right)^{2}\sigma _{X}^{2}-{\frac {1}{4}}\left(f''(\mu _{X})\right)^{2}\sigma _{X}^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94fb3c55d082462f42a9a9b9cb7bae4d1e7c3053)
![{\displaystyle \operatorname {E} \left[Y^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/903a9bbb4f14aa8ecd2db07dd1180baa01d6b30f)
![{\displaystyle \operatorname {var} \left[Y\right]=\operatorname {E} \left[Y^{2}\right]-\mu _{Y}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93ec1a80c2e68b3f7b91aaffd5c0655600c10a4f)
![{\displaystyle \operatorname {var} \left[{\frac {X}{Y}}\right]\approx {\frac {\operatorname {var} \left[X\right]}{\operatorname {E} \left[Y\right]^{2}}}-{\frac {2\operatorname {E} \left[X\right]}{\operatorname {E} \left[Y\right]^{3}}}\operatorname {cov} \left[X,Y\right]+{\frac {\operatorname {E} \left[X\right]^{2}}{\operatorname {E} \left[Y\right]^{4}}}\operatorname {var} \left[Y\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/011aff1036d96609635f44161e05afa36d783d19)
![{\displaystyle \operatorname {var} \left[f(X)\right]\approx \left(f'(\operatorname {E} \left[X\right])\right)^{2}\operatorname {var} \left[X\right]+{\frac {\left(f''(\operatorname {E} \left[X\right])\right)^{2}}{2}}\left(\operatorname {var} \left[X\right]\right)^{2}=\left(f'(\mu _{X})\right)^{2}\sigma _{X}^{2}+{\frac {1}{2}}\left(f''(\mu _{X})\right)^{2}\sigma _{X}^{4}+\left(f'(\mu _{X})\right)\left(f'''(\mu _{X})\right)\sigma _{X}^{4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c6f8695bf69e9302602a388257f566ba6f1891)
![{\displaystyle \operatorname {cov} \left[f(X),f(Y)\right]=\operatorname {E} \left[f(X)f(Y)\right]-\operatorname {E} \left[f(X)\right]\operatorname {E} \left[f(Y)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb30d1f4b08217ad7dc8430b0af14b93f0a27a02)
![{\displaystyle \operatorname {E} \left[f(X)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/904f0b10ffff22db15df5d7556258e70aeaa9693)
![{\displaystyle \operatorname {E} \left[f(X)f(Y)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/306d2c5c194290007ebe5a25f48e47107ae5a61d)
![{\displaystyle {\begin{aligned}f(X)f(Y)&{}\approx f(\mu _{X})f(\mu _{Y})+(X-\mu _{X})f'(\mu _{X})f(\mu _{Y})+(Y-\mu _{Y})f(\mu _{X})f'(\mu _{Y})+{\frac {1}{2}}\left[(X-\mu _{X})^{2}f''(\mu _{X})f(\mu _{Y})+2(X-\mu _{X})(Y-\mu _{Y})f'(\mu _{X})f'(\mu _{Y})+(Y-\mu _{Y})^{2}f(\mu _{X})f''(\mu _{Y})\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1967664e3eb99df8bca7a26321484a0cbdeb07f)
![{\displaystyle \operatorname {E} (X^{2})=\operatorname {var} (X)+\left[\operatorname {E} (X)\right]^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/787ab52bc02aee12a354b05e722f54c59e78e48e)
![{\displaystyle \operatorname {E} (XY)=\operatorname {cov} (X,Y)+\left[\operatorname {E} (X)\right]\left[\operatorname {E} (Y)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dcf130a425a5c5c545cdecd7ecbda55eaff943d)
![{\displaystyle \operatorname {E} \left[f(X)f(Y)\right]\approx f(\mu _{X})f(\mu _{Y})+f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)+{\frac {1}{2}}f''(\mu _{X})f(\mu _{Y})\operatorname {var} (X)+{\frac {1}{2}}f(\mu _{X})f''(\mu _{Y})\operatorname {var} (Y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/902dd8ed2792705004ada3a57b5fe1fa288b8b76)
![{\displaystyle {\begin{aligned}\operatorname {cov} \left[f(X),f(Y)\right]&{}\approx f(\mu _{X})f(\mu _{Y})+f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)+{\frac {1}{2}}f''(\mu _{X})f(\mu _{Y})\operatorname {var} (X)+{\frac {1}{2}}f(\mu _{X})f''(\mu _{Y})\operatorname {var} (Y)-\left[f(\mu _{X})+{\frac {1}{2}}f''(\mu _{X})\operatorname {var} (X)\right]\left[f(\mu _{Y})+{\frac {1}{2}}f''(\mu _{Y})\operatorname {var} (Y)\right]\\&{}=f'(\mu _{X})f'(\mu _{Y})\operatorname {cov} (X,Y)-{\frac {1}{4}}f''(\mu _{X})f''(\mu _{Y})\operatorname {var} (X)\operatorname {var} (Y)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43a68317901c1d4652df64c01890265e1d49b2d4)
