|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[probability theory]], a '''conditional expectation''' (also known as '''conditional expected value''' or '''conditional mean''') is the [[expected value]] of a real [[random variable]] with respect to a [[conditional probability distribution]].
| |
|
| |
|
| The concept of conditional expectation is extremely important in [[Andrey Kolmogorov|Kolmogorov]]'s [[measure theory|measure-theoretic]] definition of [[probability theory]]. In fact, the concept of conditional probability itself is actually defined in terms of conditional expectation.
| |
|
| |
|
| == Introduction ==
| | Hi there. The [http://Www.adobe.com/cfusion/search/index.cfm?term=&author%27s&loc=en_us&siteSection=home author's] name is Dalton still it's not the the majority of masucline name out that there. The hobby for him with his kids is to actually drive and he's started doing it for a while. He will work as a [http://www.Dict.cc/englisch-deutsch/cashier.html cashier]. His wife and him chose to live on in South Carolina additionally his family loves the following. Go to his website identify out more: http://prometeu.net<br><br>my homepage; clash of clans hack - [http://prometeu.net click through the next website], |
| Let ''X'' and ''Y'' be [[discrete random variable]]s, then the '''conditional expectation''' of ''X'' given the event ''Y=y'' is a function of ''y'' over the range of ''Y''
| |
| | |
| :<math> \operatorname{E} (X | Y=y ) = \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x|Y=y) = \sum_{x \in \mathcal{X}} x \ \frac{\operatorname{P}(X=x,Y=y)}{\operatorname{P}(Y=y)}, </math> | |
| | |
| where <math>\mathcal{X}</math> is the [[Range (mathematics)|range]] of ''X''.
| |
| | |
| If now ''X'' is a [[continuous random variable]], while ''Y'' remains a discrete variable, the '''conditional expectation''' is:
| |
| :<math> \operatorname{E} (X | Y=y )= \int_{\mathcal{X}} x f_X (x |Y=y) dx </math>
| |
| | |
| where <math> f_X (\,\cdot\, |Y=y)</math> is the [[conditional density]] of <math>X</math> given <math>Y=y</math>.
| |
| | |
| A problem arises when ''Y'' is [[continuous random variable|continuous]]. In this case, the probability P(''Y=y'') = 0, and the [[Borel–Kolmogorov paradox]] demonstrates the ambiguity of attempting to define conditional probability along these lines.
| |
| | |
| However the above expression may be rearranged:
| |
| | |
| :<math> \operatorname{E} (X | Y=y) \operatorname{P}(Y=y) = \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x,Y=y), </math>
| |
| | |
| and although this is trivial for individual values of ''y'' (since both sides are zero), it should hold for any measurable subset ''B'' of the domain of ''Y'' that:
| |
| | |
| :<math> \int_B \operatorname{E} (X | Y=y) \operatorname{P}(Y=y) \ \operatorname{d}y = \int_B \sum_{x \in \mathcal{X}} x \ \operatorname{P}(X=x,Y=y) \ \operatorname{d}y. </math>
| |
| | |
| In fact, this is a sufficient condition to define both conditional expectation and conditional probability.
| |
| | |
| == Formal definition ==
| |
| Let <math>\scriptstyle (\Omega, \mathcal {F}, \operatorname {P} )</math> be a [[probability space]], with a [[random variable#Formal definition|random variable]] <math>\scriptstyle X:\Omega \to \mathbb{R}^n</math> and a sub-[[sigma-algebra|σ-algebra]] <math>\scriptstyle \mathcal {H} \subseteq \mathcal {F} </math>.
| |
| | |
| Then a '''conditional expectation''' of ''X'' given <math>\scriptstyle \mathcal {H} </math> (denoted as <math>\scriptstyle \operatorname{E}\left[X|\mathcal {H} \right]</math>) is any <math>\scriptstyle \mathcal {H} </math>-[[measurable function]] (<math>\Omega \to \mathbb{R}^n</math>) which satisfies:
| |
| :<math> \int_H \operatorname{E}\left[X|\mathcal {H} \right] (\omega) \ \operatorname{d} \operatorname{P}(\omega) = \int_H X(\omega) \ \operatorname{d} \operatorname{P}(\omega) \qquad \text{for each} \quad H \in \mathcal {H} </math>.<ref>[[#loe1978|Loève (1978)]], p. 7</ref>
| |
| | |
| Note that <math>\scriptstyle \operatorname{E}\left[X|\mathcal {H} \right]</math> is simply the name of the conditional expectation function.
| |
| | |
| === Discussion ===
| |
| A couple of points worth noting about the definition:
| |
| * This is not a constructive definition; we are merely given the required property that a conditional expectation must satisfy.
| |
| ** The required property has the same form as the last expression in the Introduction section.
| |
| ** Existence of a conditional expectation function is determined by the [[Radon–Nikodym theorem]], a sufficient condition is that the (unconditional) expected value for ''X'' exist.
| |
| ** Uniqueness can be shown to be [[almost surely|almost sure]]: that is, versions of the same conditional expectation will only differ on a [[null set|set of probability zero]].
| |
| * The σ-algebra <math>\scriptstyle \mathcal {H} </math> controls the "granularity" of the conditioning. A conditional expectation <math>\scriptstyle{E}\left[X|\mathcal {H} \right]</math> over a finer-grained σ-algebra <math>\scriptstyle \mathcal {H} </math> will allow us to condition on a wider variety of events.
| |
| ** To condition freely on values of a random variable ''Y'' with state space <math>\scriptstyle (\mathcal Y, \Sigma) </math>, it suffices to define the conditional expectation using the [[pre-image]] of ''Σ'' with respect to ''Y'', so that <math>\scriptstyle \operatorname{E}\left[X| Y\right]</math> is defined to be <math>\scriptstyle \operatorname{E}\left[X|\mathcal {H} \right]</math>, where
| |
| :::<math> \mathcal {H} = \sigma(Y):= Y^{-1}\left(\Sigma\right):= \{Y^{-1}(S) : S \in \Sigma \} </math>
| |
| :: This suffices to ensure that the conditional expectation is σ(''Y'')-measurable. Although conditional expectation is defined to condition on events in the underlying probability space Ω, the requirement that it be σ(''Y'')-measurable allows us to condition on ''Y'' as in the introduction.
| |
| | |
| == Definition of conditional probability ==
| |
| For any event <math>A \in \mathcal{A} \supseteq \mathcal B</math>, define the [[indicator function]]:
| |
| | |
| :<math>\mathbf{1}_A (\omega) = \begin{cases} 1 \; &\text{if } \omega \in A, \\ 0 \; &\text{if } \omega \notin A, \end{cases}</math>
| |
| | |
| which is a random variable with respect to the [[Borel algebra|Borel σ-algebra]] on (0,1). Note that the expectation of this random variable is equal to the probability of ''A'' itself:
| |
| | |
| :<math>\operatorname{E}(\mathbf{1}_A) = \operatorname{P}(A). \; </math>
| |
| | |
| Then the '''[[conditional probability]] given <math>\scriptstyle \mathcal B</math>''' is a function <math>\scriptstyle \operatorname{P}(\cdot|\mathcal{B}):\mathcal{A} \times \Omega \to (0,1)</math> such that <math>\scriptstyle \operatorname{P}(A|\mathcal{B})</math> is the conditional expectation of the indicator function for ''A'':
| |
| | |
| :<math>\operatorname{P}(A|\mathcal{B}) = \operatorname{E}(\mathbf{1}_A|\mathcal{B}) \; </math>
| |
| | |
| In other words, <math>\scriptstyle \operatorname{P}(A|\mathcal{B}) </math> is a <math>\scriptstyle \mathcal B</math>-measurable function satisfying
| |
| | |
| :<math>\int_B \operatorname{P}(A|\mathcal{B}) (\omega) \, \operatorname{d} \operatorname{P}(\omega) = \operatorname{P} (A \cap B) \qquad \text{for all} \quad A \in \mathcal{A}, B \in \mathcal{B}. </math>
| |
| | |
| A conditional probability is [[Regular_conditional_probability|'''regular''']] if <math>\scriptstyle \operatorname{P}(\cdot|\mathcal{B})(\omega) </math> is also a [[probability measure]] for all ''ω'' ∈ ''Ω''. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.
| |
| | |
| * For the trivial sigma algebra <math>\mathcal B= \{\emptyset,\Omega\}</math> the conditional probability is a constant function, <math>\operatorname{P}\!\left( A| \{\emptyset,\Omega\} \right) \equiv\operatorname{P}(A).</math>
| |
| | |
| * For <math>A\in \mathcal{B}</math>, as outlined above, <math>\operatorname{P}(A|\mathcal{B})=1_A.</math>.
| |
| | |
| See also [[Conditional_probability_distribution#Measure-Theoretic_Formulation|conditional probability distribution]].
| |
| | |
| == Conditioning as factorization ==
| |
| | |
| In the definition of conditional expectation that we provided above, the fact that ''Y'' is a ''real'' random variable is irrelevant: Let ''U'' be a measurable space, that is, a set equipped with a σ-algebra <math>\Sigma</math> of subsets. A ''U''-valued random variable is a function <math>Y\colon (\Omega,\mathcal A) \mapsto (U,\Sigma)</math> such that <math>Y^{-1}(B)\in \mathcal A</math> for any measurable subset <math>B\in \Sigma</math> of ''U''.
| |
| | |
| We consider the measure Q on ''U'' given as above: Q(''B'') = P(''Y''<sup>−1</sup>(''B'')) for every measurable subset ''B'' of ''U''. Then Q is a probability measure on the measurable space ''U'' defined on its σ-algebra of measurable sets.
| |
| | |
| '''Theorem'''. If ''X'' is an integrable random variable on Ω then there is one and, up to equivalence a.e. relative to Q, only one integrable function ''g'' on ''U'' (which is written <math>g= \operatorname{E}(X \mid Y)</math>) such that for any measurable subset ''B'' of ''U'':
| |
| | |
| :<math> \int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} g(u) \ d \operatorname{Q} (u). </math>
| |
| | |
| There are a number of ways of proving this; one as suggested above, is to note that the expression on the left hand side defines, as a function of the set ''B'', a countably additive signed measure ''μ'' on the measurable subsets of ''U''. Moreover, this measure ''μ'' is absolutely continuous relative to Q. Indeed Q(''B'') = 0 means exactly that ''Y''<sup>−1</sup>(''B'') has probability 0. The integral of an integrable function on a set of probability 0 is itself 0. This proves absolute continuity. Then the [[Radon–Nikodym theorem]] provides the function ''g'', equal to the density of ''μ'' with respect to Q.
| |
| | |
| The defining condition of conditional expectation then is the equation
| |
| | |
| :<math> \int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} \operatorname{E}(X \mid Y)(u) \ d \operatorname{Q} (u),</math>
| |
| and it holds that | |
| :<math>\operatorname{E}(X \mid Y) \circ Y= \operatorname{E}\left(X \mid Y^{-1} \left(\Sigma\right)\right).</math>
| |
| | |
| We can further interpret this equality by considering the abstract [[change of variables]] formula to transport the integral on the right hand side to an integral over Ω:
| |
| | |
| :<math> \int_{Y^{-1}(B)} X(\omega) \ d \operatorname{P}(\omega) = \int_{Y^{-1}(B)} (\operatorname{E}(X \mid Y) \circ Y)(\omega) \ d \operatorname{P} (\omega). </math>
| |
| | |
| This equation can be interpreted to say that the following diagram is [[commutative diagram|commutative]] ''in the average''.
| |
| | |
| <!-- Picture is wrong
| |
| :[[Image:Conditional expectation commutative diagram.png|200px|left|A diagram, commutative in an average sense.]] | |
| <br style="clear:left" />
| |
| -->
| |
| | |
| E(X|Y)= goY
| |
| Ω ───────────────────────────> '''R'''
| |
| Y g=E(X|Y= ·)
| |
| Ω ──────────> '''R''' ───────────> '''R'''
| |
|
| |
| ω ──────────> Y(ω) ───────────> g(Y(ω)) = E(X|Y=Y(ω))
| |
|
| |
| y ───────────> g( y ) = E(X|Y= y )
| |
| | |
| The equation means that the integrals of ''X'' and the composition <math>\operatorname{E}(X \mid Y=\ \cdot)\circ Y</math> over sets of the form ''Y''<sup>−1</sup>(''B''), for ''B'' a measurable subset of ''U'', are identical.
| |
| | |
| == Conditioning relative to a subalgebra ==
| |
| | |
| There is another viewpoint for conditioning involving σ-subalgebras ''N'' of the σ-algebra ''M''. This version is a trivial specialization of the preceding: we simply take ''U'' to be the space Ω with the σ-algebra ''N'' and ''Y'' the identity map. We state the result:
| |
| | |
| '''Theorem'''. If ''X'' is an integrable real random variable on Ω then there is one and, up to equivalence a.e. relative to P, only one integrable function ''g'' such that for any set ''B'' belonging to the subalgebra ''N''
| |
| | |
| :<math> \int_{B} X(\omega) \ d \operatorname{P}(\omega) = \int_{B} g(\omega) \ d \operatorname{P} (\omega) </math>
| |
| | |
| where ''g'' is measurable with respect to ''N'' (a stricter condition than the measurability with
| |
| respect to ''M'' required of ''X'').
| |
| This form of conditional expectation is usually written: E(''X'' | ''N'').
| |
| This version is preferred by probabilists. One reason is that on the [[Hilbert space]] of [[square-integrable]] real random variables (in other words, real random variables with finite second moment) the mapping ''X'' → E(''X'' | ''N'')
| |
| is [[self-adjoint operator|self-adjoint]]
| |
| | |
| :<math>\operatorname E(X\cdot\operatorname E(Y\mid N)) = \operatorname E\left(\operatorname E(X\mid N)\cdot \operatorname E(Y\mid N)\right) = \operatorname E(\operatorname E(X\mid N)\cdot Y)</math> | |
| | |
| and a [[projection]] (i.e. idempotent)
| |
| :<math> L^2_{\operatorname{P}}(\Omega;M) \rightarrow L^2_{\operatorname{P}}(\Omega;N). </math>
| |
| | |
| == Basic properties ==
| |
| | |
| Let (Ω, ''M'', P) be a probability space, and let ''N'' be a σ-subalgebra of ''M''.
| |
| | |
| * Conditioning with respect to ''N''  is linear on the space of integrable real random variables.
| |
| | |
| * <math>\operatorname{E}(1\mid N) = 1. </math> More generally, <math>\operatorname{E} (Y\mid N)= Y</math> for every integrable ''N''–measurable random variable ''Y'' on Ω.
| |
| | |
| * <math>\operatorname{E}(1_B \,\operatorname{E} (X\mid N))= \operatorname{E}(1_B \, X)</math>   for all ''B'' ∈ ''N'' and every integrable random variable ''X'' on Ω.
| |
| | |
| * [[Jensen's inequality]] holds: If ''ƒ'' is a [[convex function]], then
| |
| | |
| :: <math> f(\operatorname{E}(X \mid N) ) \leq \operatorname{E}(f \circ X \mid N).</math>
| |
| | |
| * Conditioning is a contractive projection
| |
| | |
| ::<math> L^s_P(\Omega; M) \rightarrow L^s_P(\Omega; N), \text{ i.e. } \operatorname{E}|\operatorname{E}(X\mid N)|^s \le \operatorname{E}|X|^s</math>
| |
| :for any ''s'' ≥ 1.
| |
| | |
| ==See also==
| |
| *[[Law of total probability]]
| |
| *[[Law of total expectation]]
| |
| *[[Law of total variance]]
| |
| *[[Law of total cumulance]] (generalizes the other three)
| |
| *[[Conditioning (probability)]]
| |
| *[[Joint probability distribution]]
| |
| *[[Disintegration theorem]]
| |
| | |
| == Notes ==
| |
| {{Reflist}}
| |
| | |
| {{More footnotes|date=November 2010}}
| |
| | |
| == References ==
| |
| {{Refbegin}}
| |
| * {{Cite book |title= Grundbegriffe der Wahrscheinlichkeitsrechnung |last= Kolmogorov |first= Andrey |authorlink= Andrey Kolmogorov |coauthors= |year= 1933 |publisher= Julius Springer |location= Berlin |isbn= |page= |pages= |url= |language= German |ref= kol1933}}{{Page needed|date=December 2010}}
| |
| ** Translation: {{cite book |title= Foundations of the Theory of Probability |edition= 2nd |last= Kolmogorov |first= Andrey |authorlink= Andrey Kolmogorov |coauthors= |year= 1956 |publisher= Chelsea |location= New York |isbn= 0-8284-0023-7 |page= |pages= |url= http://www.mathematik.com/Kolmogorov/index.html |ref= kol1956 }}{{Page needed|date=December 2010}}
| |
| * {{Cite book|last=Loève|first=Michel|authorlink=Michel Loève|title=Probability Theory vol. II |edition=4th |publisher=Springer |year=1978 |isbn=0-387-90262-7 |chapter=Chapter 27. Concept of Conditioning|ref=loe1978}}{{Page needed|date=December 2010}}
| |
| * [[William Feller]], ''An Introduction to Probability Theory and its Applications'', vol 1, 1950 {{Page needed|date=December 2010}}
| |
| * Paul A. Meyer, ''Probability and Potentials'', Blaisdell Publishing Co., 1966 {{Page needed|date=December 2010}}
| |
| * {{Cite book|last1=Grimmett|first1=Geoffrey|authorlink=Geoffrey Grimmett |last2=Stirzaker|first2=David|title=Probability and Random Processes|year=2001|edition=3rd|publisher=Oxford University Press|isbn=0-19-857222-0}}, pages 67-69
| |
| {{Refend}}
| |
| | |
| == External links ==
| |
| * {{Springer |title=Conditional mathematical expectation |id=c/c024500 |first=N.G. |last=Ushakov }}
| |
| | |
| {{DEFAULTSORT:Conditional Expectation}}
| |
| [[Category:Probability theory]]
| |
| [[Category:Statistical terminology]]
| |