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The density of a [[Conditional distribution|conditional distribution]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244701.png" /> be a probability space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244702.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244703.png" />-algebra of Borel sets on the line, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244704.png" /> be a sub-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244705.png" />-algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244706.png" />, let
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244707.png" /></td> </tr></table>
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be the conditional distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244708.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c0244709.png" />, and let
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The density of a [[Conditional distribution|conditional distribution]]. Let  $  ( \Omega , {\mathcal A} , {\mathsf P} ) $
 +
be a probability space, let  $  \mathfrak B $
 +
be the $  \sigma $-
 +
algebra of Borel sets on the line, let  $  \mathfrak F $
 +
be a sub- $  \sigma $-
 +
algebra of $  {\mathcal A} $,  
 +
let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447010.png" /></td> </tr></table>
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$$
 +
Q ( \omega , B )  = \
 +
{\mathsf P} \{ X \in B  \mid  \mathfrak F  \} ,\ \
 +
\omega \in \Omega ,\ \
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B \in \mathfrak B ,
 +
$$
  
be the conditional distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447012.png" />. If
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be the conditional distribution of $  X $
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with respect to $  \mathfrak F $,
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and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447013.png" /></td> </tr></table>
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$$
 +
F _ {X} ( x \mid  \mathfrak F  ) \
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= Q ( \omega , ( - \infty , x ) )
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447014.png" /> is called the conditional density of the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447015.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447016.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447017.png" />.
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be the conditional distribution function of $  X $
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with respect to $  \mathfrak F $.  
 +
If
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447019.png" /> are random variables, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447020.png" /> is the density of the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447022.png" /> is the joint density of the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447024.png" />, then
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$$
 +
F _ {X} ( x \mid  \mathfrak F  )  = \
 +
\int\limits _ {- \infty } ^ { x }
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f _ {X} ( t \mid  \mathfrak F  )  d t ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447025.png" /></td> </tr></table>
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then  $  f _ {X} ( x \mid  \mathfrak F  ) $
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is called the conditional density of the distribution of  $  X $
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with respect to the  $  \sigma $-
 +
algebra  $  \mathfrak F $.
  
defines the conditional density of the distribution of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447026.png" /> for fixed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024470/c02447029.png" />.
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If  $  X $
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and  $  Y $
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are random variables,  $  f _ {Y} ( y) $
 +
is the density of the distribution of  $  Y $
 +
and  $  f _ {X,Y} ( x , y ) $
 +
is the joint density of the distribution of  $  X $
 +
and  $  Y $,
 +
then
 +
 
 +
$$
 +
f _ {X} ( x \mid  Y = y )  = \
 +
 
 +
\frac{1}{f _ {Y} ( y) }
 +
 
 +
f _ {X,Y} ( x , y )
 +
$$
 +
 
 +
defines the conditional density of the distribution of the random variable $  X $
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for fixed values $  y $
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of $  Y $
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for which $  f _ {Y} ( y) \neq 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. [Yu.V. Prokhorov] Prohorov,  Yu.A. Rozanov,  "Probability theory, basic concepts. Limit theorems, random processes" , Springer  (1969)  (Translated from Russian)</TD></TR></table>

Latest revision as of 17:46, 4 June 2020


The density of a conditional distribution. Let $ ( \Omega , {\mathcal A} , {\mathsf P} ) $ be a probability space, let $ \mathfrak B $ be the $ \sigma $- algebra of Borel sets on the line, let $ \mathfrak F $ be a sub- $ \sigma $- algebra of $ {\mathcal A} $, let

$$ Q ( \omega , B ) = \ {\mathsf P} \{ X \in B \mid \mathfrak F \} ,\ \ \omega \in \Omega ,\ \ B \in \mathfrak B , $$

be the conditional distribution of $ X $ with respect to $ \mathfrak F $, and let

$$ F _ {X} ( x \mid \mathfrak F ) \ = Q ( \omega , ( - \infty , x ) ) $$

be the conditional distribution function of $ X $ with respect to $ \mathfrak F $. If

$$ F _ {X} ( x \mid \mathfrak F ) = \ \int\limits _ {- \infty } ^ { x } f _ {X} ( t \mid \mathfrak F ) d t , $$

then $ f _ {X} ( x \mid \mathfrak F ) $ is called the conditional density of the distribution of $ X $ with respect to the $ \sigma $- algebra $ \mathfrak F $.

If $ X $ and $ Y $ are random variables, $ f _ {Y} ( y) $ is the density of the distribution of $ Y $ and $ f _ {X,Y} ( x , y ) $ is the joint density of the distribution of $ X $ and $ Y $, then

$$ f _ {X} ( x \mid Y = y ) = \ \frac{1}{f _ {Y} ( y) } f _ {X,Y} ( x , y ) $$

defines the conditional density of the distribution of the random variable $ X $ for fixed values $ y $ of $ Y $ for which $ f _ {Y} ( y) \neq 0 $.

References

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
How to Cite This Entry:
Conditional density. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conditional_density&oldid=15893
This article was adapted from an original article by V.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article