Conditional variance: Difference between revisions

Revision as of 10:40, 8 October 2022

Latest revision as of 10:03, 4 June 2024

(7 intermediate revisions by 6 users not shown)

Line 1:

{{Short description|Variance of a random variable given value of other variables}}

In [[probability theory]] and [[statistics]], a '''conditional variance''' is the [[variance]] of a [[random variable]] given the value(s) of one or more other variables.

Particularly in [[econometrics]], the conditional variance is also known as the '''scedastic function''' or '''skedastic function'''.<ref>{{cite book |first=Aris |last=Spanos |chapter=Conditioning and regression |title=Probability Theory and Statistical Inference |___location=New York |publisher=Cambridge University Press |year=1999 |isbn=0-521-42408-9 |pages=339–356 [p. 342] |url=https://books.google.com/books?id=G0_HxBubGAwC&pg=PA342 }}</ref> Conditional variances are important parts of [[autoregressive conditional heteroskedasticity]] (ARCH) models.

Line 5 ⟶ 6:

The conditional variance of a [[random variable]] ''Y'' given another random variable ''X'' is

:<math>\operatorname{Var}(Y|\mid X) = \operatorname{E}\Big(\big(Y - \operatorname{E}(Y\mid X)\big)^{2}\mid;\Big|\; X\Big).</math>

The conditional variance tells us how much variance is left if we use <math>\operatorname{E}(Y\mid X)</math> to "predict" ''Y''.

Here, as usual, <math>\operatorname{E}(Y\mid X)</math> stands for the [[conditional expectation]] of ''Y'' given ''X'',

which we may recall, is a random variable itself (a function of ''X'', determined up to probability one).

As a result, <math>\operatorname{Var}(Y|\mid X)</math> itself is a random variable (and is a function of ''X'').

==Explanation, relation to least-squares ==

Line 32 ⟶ 33:

==Special cases, variations==

===Conditioning on discrete random variables===

When ''X'' takes on countable many values <math>S = \{x_1,x_1x_2,\dots\}</math> with positive probability, i.e., it is a [[discrete random variable]], we can introduce <math>\operatorname{Var}(Y|X=x)</math>, the conditional variance of ''Y'' given that ''X=x'' for any ''x'' from ''S'' as follows:

:<math>\operatorname{Var}(Y|X=x) = \operatorname{E}((Y - \operatorname{E}(Y\mid X=x))^{2}\mid X=x)=\operatorname{E}(Y^2|X=x)-\operatorname{E}(Y|X=x)^2,</math>

where recall that <math>\operatorname{E}(Z\mid X=x)</math> is the [[Conditional_expectation#Conditional_expectation_with_respect_to_a_random_variable|conditional expectation of ''Z'' given that ''X=x'']], which is well-defined for <math>x\in S</math>.

Line 74 ⟶ 75:

[[Category:Theory of probability distributions]]

[[Category:Conditional probability]]

{{statistics-stub}}