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true |
Joram Soch |
BCCN Berlin |
joram.soch@bccn-berlin.de |
2021-11-26 03:20:00 -0800 |
Law of total variance |
General Theorems |
Probability theory |
Variance |
Law of total variance |
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Wikipedia |
2021 |
Law of total variance |
Wikipedia, the free encyclopedia |
retrieved on 2021-11-26 |
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P292 |
var-tot |
JoramSoch |
Theorem: (law of total variance, also called "conditional variance formula") Let $X$ and $Y$ be random variables defined on the same probability space and assume that the variance of $Y$ is finite. Then, the sum of the expectation of the conditional variance and the variance of the conditional expectation of $Y$ given $X$ is equal to the variance of $Y$:
$$ \label{eq:var-tot}
\mathrm{Var}(Y) = \mathrm{E}[\mathrm{Var}(Y \vert X)] + \mathrm{Var}[\mathrm{E}(Y \vert X)] ; .
$$
Proof: The variance can be decomposed into expected values as follows:
$$ \label{eq:var-tot-s1}
\mathrm{Var}(Y) = \mathrm{E}(Y^2) - \mathrm{E}(Y)^2 ; .
$$
This can be rearranged into:
$$ \label{eq:var-tot-s2}
\mathrm{E}(Y^2) = \mathrm{Var}(Y) + \mathrm{E}(Y)^2 ; .
$$
Applying the law of total expectation, we have:
$$ \label{eq:var-tot-s3}
\mathrm{E}(Y^2) = \mathrm{E}\left[ \mathrm{Var}(Y \vert X) + \mathrm{E}(Y \vert X)^2 \right] ; .
$$
Now subtract the second term from \eqref{eq:var-tot-s1}:
$$ \label{eq:var-tot-s4}
\mathrm{E}(Y^2) - \mathrm{E}(Y)^2 = \mathrm{E}\left[ \mathrm{Var}(Y \vert X) + \mathrm{E}(Y \vert X)^2 \right] - \mathrm{E}(Y)^2 ; .
$$
Again applying the law of total expectation, we have:
$$ \label{eq:var-tot-s5}
\mathrm{E}(Y^2) - \mathrm{E}(Y)^2 = \mathrm{E}\left[ \mathrm{Var}(Y \vert X) + \mathrm{E}(Y \vert X)^2 \right] - \mathrm{E}\left[ \mathrm{E}(Y \vert X) \right]^2 ; .
$$
With the linearity of the expected value, the terms can be regrouped to give:
$$ \label{eq:var-tot-s6}
\mathrm{E}(Y^2) - \mathrm{E}(Y)^2 = \mathrm{E}\left[ \mathrm{Var}(Y \vert X) \right] + \left( \mathrm{E}\left[ \mathrm{E}(Y \vert X)^2 \right] - \mathrm{E}\left[ \mathrm{E}(Y \vert X) \right]^2 \right) ; .
$$
Using the decomposition of variance into expected values, we finally have:
$$ \label{eq:var-tot-s7}
\mathrm{Var}(Y) = \mathrm{E}[\mathrm{Var}(Y \vert X)] + \mathrm{Var}[\mathrm{E}(Y \vert X)] ; .
$$