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''probability density''
 
''probability density''
  
 
The derivative of the [[Distribution function|distribution function]] corresponding to an absolutely-continuous probability measure.
 
The derivative of the [[Distribution function|distribution function]] corresponding to an absolutely-continuous probability measure.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311101.png" /> be a random vector taking values in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311102.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311103.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311104.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311105.png" /> be its distribution function, and let there exist a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311106.png" /> such that
+
Let $  X $
 +
be a random vector taking values in an $  n $-
 +
dimensional Euclidean space $  \mathbf R  ^ {n} $
 +
$  ( n \geq  1) $,  
 +
let $  F $
 +
be its distribution function, and let there exist a non-negative function $  f $
 +
such that
  
<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/d/d031/d031110/d0311107.png" /></td> </tr></table>
+
$$
 +
F( x _ {1} \dots x _ {n} )  = \int\limits _ { - \infty } ^ { {x _ 1 } } \dots \int\limits _ { - \infty } ^ { {x _ n } } f( u _ {1} \dots u _ {n} )  du _ {1} \dots du _ {n}  $$
  
for any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311108.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d0311109.png" /> is called the probability density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111010.png" />, and for any Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111011.png" />,
+
for any real $  x _ {1} \dots x _ {n} $.  
 +
Then $  f $
 +
is called the probability density of $  X $,  
 +
and for any Borel set $  A\subset  \mathbf R  ^ {n} $,
  
<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/d/d031/d031110/d03111012.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} \{ X \in A \}  = {\int\limits \dots \int\limits } _ { A } f( u _ {1} \dots u _ {n} )  du _ {1} {} \dots du _ {n} .
 +
$$
  
Any non-negative integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111013.png" /> satisfy the condition
+
Any non-negative integrable function $  f $
 +
satisfy the condition
  
<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/d/d031/d031110/d03111014.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty  f( x _ {1} \dots x _ {n} )  dx _ {1} \dots dx _ {n}  = 1
 +
$$
  
 
is the probability density of some random vector.
 
is the probability density of some random vector.
  
If two random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111016.png" /> taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111017.png" /> are independent and have probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111019.png" /> respectively, then the random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111020.png" /> has the probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111021.png" /> that is the convolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111023.png" />:
+
If two random vectors $  X $
 +
and $  Y $
 +
taking values in $  \mathbf R  ^ {n} $
 +
are independent and have probability densities $  f $
 +
and $  g $
 +
respectively, then the random vector $  X+ Y $
 +
has the probability density $  h $
 +
that is the convolution of $  f $
 +
and  $  g $:
 +
 
 +
$$
 +
h( x _ {1} \dots x _ {n} ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty 
 +
f( x _ {1} - u _ {1} \dots x _ {n} - u _ {n} ) g( u _ {1} \dots u _ {n} ) \times
 +
$$
  
<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/d/d031/d031110/d03111024.png" /></td> </tr></table>
+
$$
 +
\times
 +
du _ {1} \dots du _ {n\ } =
 +
$$
  
<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/d/d031/d031110/d03111025.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty  f( u _ {1} \dots
 +
u _ {n} ) g( x _ {1} - u _ {1} \dots x _ {n} - u _ {n} ) \times
 +
$$
  
<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/d/d031/d031110/d03111026.png" /></td> </tr></table>
+
$$
 +
\times \
 +
du _ {1} \dots du _ {n} .
 +
$$
  
<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/d/d031/d031110/d03111027.png" /></td> </tr></table>
+
Let  $  X = ( X _ {1} \dots X _ {n} ) $
 +
and  $  Y = ( Y _ {1} \dots Y _ {m} ) $
 +
be random vectors taking values in  $  \mathbf R  ^ {n} $
 +
and  $  \mathbf R  ^ {m} $
 +
$  ( n, m \geq  1) $
 +
and having probability densities  $  f $
 +
and  $  g $
 +
respectively, and let  $  Z = ( X _ {1} \dots X _ {n} , Y _ {1} \dots Y _ {m} ) $
 +
be a random vector in  $  \mathbf R  ^ {n+} m $.  
 +
If then  $  X $
 +
and  $  Y $
 +
are independent,  $  Z $
 +
has the probability density  $  h $,
 +
which is called the joint probability density of the random vectors  $  X $
 +
and  $  Y $,
 +
where
  
<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/d/d031/d031110/d03111028.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
h( t _ {1} \dots t _ {n+} m )  = f( t _ {1} \dots t _ {n} ) g( t _ {n+} 1 \dots t _ {n+} m ).
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111030.png" /> be random vectors taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111032.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111033.png" /> and having probability densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111035.png" /> respectively, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111036.png" /> be a random vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111037.png" />. If then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111039.png" /> are independent, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111040.png" /> has the probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111041.png" />, which is called the joint probability density of the random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111043.png" />, where
+
Conversely, if  $  Z $
 +
has a probability density that satisfies (1), then $  X $
 +
and $  Y $
 +
are independent.
  
<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/d/d031/d031110/d03111044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
The characteristic function  $  \phi $
 +
of a random vector  $  X $
 +
having a probability density  $  f $
 +
is expressed by
  
Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111045.png" /> has a probability density that satisfies (1), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111047.png" /> are independent.
+
$$
 +
\phi ( t _ {1} \dots t _ {n} ) =
 +
$$
  
The characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111048.png" /> of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111049.png" /> having a probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111050.png" /> is expressed by
+
$$
 +
= \
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty  e ^ {i( t _ {1} x _ {1} + \dots + t _ {n} x _ {n} ) } f( x _ {1} \dots x _ {n} )  dx _ {1} \dots dx _ {n} ,
 +
$$
  
<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/d/d031/d031110/d03111051.png" /></td> </tr></table>
+
where if  $  \phi $
 +
is absolutely integrable then  $  f $
 +
is a bounded continuous function, and
  
<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/d/d031/d031110/d03111052.png" /></td> </tr></table>
+
$$
 +
f( x _ {1} \dots x _ {n} ) =
 +
$$
  
where if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111053.png" /> is absolutely integrable then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111054.png" /> is a bounded continuous function, and
+
$$
 +
= \
  
<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/d/d031/d031110/d03111055.png" /></td> </tr></table>
+
\frac{1}{( 2 \pi )  ^ {n} }
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty
 +
} ^  \infty  e ^ {- i( t _ {1} x _ {1} + \dots + t _ {n} x _ {n} ) } \phi ( t _ {1} \dots t _ {n} )  dt _ {1} \dots dt _ {n} .
 +
$$
  
<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/d/d031/d031110/d03111056.png" /></td> </tr></table>
+
The probability density  $  f $
 +
and the corresponding characteristic function  $  \phi $
 +
are related also by the following relation (Plancherel's identity): The function  $  f ^ { 2 } $
 +
is integrable if and only if the function  $  | \phi |  ^ {2} $
 +
is integrable, and in that case
  
The probability density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111057.png" /> and the corresponding characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111058.png" /> are related also by the following relation (Plancherel's identity): The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111059.png" /> is integrable if and only if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111060.png" /> is integrable, and in that case
+
$$
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty  f ^ { 2 } ( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n\ } =
 +
$$
  
<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/d/d031/d031110/d03111061.png" /></td> </tr></table>
+
$$
 +
= \
  
<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/d/d031/d031110/d03111062.png" /></td> </tr></table>
+
\frac{1}{( 2 \pi )  ^ {n} }
 +
\int\limits _ {- \infty } ^  \infty  \dots \int\limits _ {- \infty } ^  \infty  | \phi ( t _ {1} \dots t _ {n} ) |  ^ {2}  dt _ {1} \dots dt _ {n} .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111063.png" /> be a [[Measurable space|measurable space]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111065.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111066.png" />-finite measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111067.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111068.png" /> absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111069.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111070.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111072.png" />. In that case there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111073.png" /> a non-negative measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111074.png" /> such that
+
Let $  ( \Omega , \mathfrak A) $
 +
be a [[Measurable space|measurable space]], and let $  \nu $
 +
and $  \mu $
 +
be $  \sigma $-
 +
finite measures on $  ( \Omega , \mathfrak A) $
 +
with $  \nu $
 +
absolutely continuous with respect to $  \mu $,  
 +
i.e. $  \mu ( A) = 0 $
 +
implies $  \nu ( A) = 0 $,  
 +
$  A \in \mathfrak A $.  
 +
In that case there exists on $  ( \Omega , \mathfrak A) $
 +
a non-negative measurable function $  f $
 +
such that
  
<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/d/d031/d031110/d03111075.png" /></td> </tr></table>
+
$$
 +
\nu ( A)  = \int\limits _ { A } f  d \mu
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111076.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111077.png" /> is called the Radon–Nikodým derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111078.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111079.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111080.png" /> is a probability measure, it is also the probability density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111081.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111082.png" />.
+
for any $  A \in \mathfrak A $.  
 +
The function $  f $
 +
is called the Radon–Nikodým derivative of $  \nu $
 +
with respect to $  \mu $,  
 +
while if $  \nu $
 +
is a probability measure, it is also the probability density of $  \nu $
 +
relative to $  \mu $.
  
A concept closely related to the probability density is that of a dominated family of distributions. A family of probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111083.png" /> on a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111084.png" /> is called dominated if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111085.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111086.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111087.png" /> such that each probability measure from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111088.png" /> has a probability density relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111089.png" /> (or, what is the same, if each measure from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111090.png" /> is absolutely continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031110/d03111091.png" />). The assumption of dominance is important in certain theorems in mathematical statistics.
+
A concept closely related to the probability density is that of a dominated family of distributions. A family of probability distributions $  \mathfrak P $
 +
on a measurable space $  ( \Omega , \mathfrak A) $
 +
is called dominated if there exists a $  \sigma $-
 +
finite measure $  \mu $
 +
on $  ( \Omega , \mathfrak A) $
 +
such that each probability measure from $  \mathfrak P $
 +
has a probability density relative to $  \mu $(
 +
or, what is the same, if each measure from $  \mathfrak P $
 +
is absolutely continuous with respect to $  \mu $).  
 +
The assumption of dominance is important in certain theorems in mathematical statistics.
  
 
====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><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses", Wiley (1986)</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><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses", Wiley (1986)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


probability density

The derivative of the distribution function corresponding to an absolutely-continuous probability measure.

Let $ X $ be a random vector taking values in an $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $ $ ( n \geq 1) $, let $ F $ be its distribution function, and let there exist a non-negative function $ f $ such that

$$ F( x _ {1} \dots x _ {n} ) = \int\limits _ { - \infty } ^ { {x _ 1 } } \dots \int\limits _ { - \infty } ^ { {x _ n } } f( u _ {1} \dots u _ {n} ) du _ {1} \dots du _ {n} $$

for any real $ x _ {1} \dots x _ {n} $. Then $ f $ is called the probability density of $ X $, and for any Borel set $ A\subset \mathbf R ^ {n} $,

$$ {\mathsf P} \{ X \in A \} = {\int\limits \dots \int\limits } _ { A } f( u _ {1} \dots u _ {n} ) du _ {1} {} \dots du _ {n} . $$

Any non-negative integrable function $ f $ satisfy the condition

$$ \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty f( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n} = 1 $$

is the probability density of some random vector.

If two random vectors $ X $ and $ Y $ taking values in $ \mathbf R ^ {n} $ are independent and have probability densities $ f $ and $ g $ respectively, then the random vector $ X+ Y $ has the probability density $ h $ that is the convolution of $ f $ and $ g $:

$$ h( x _ {1} \dots x _ {n} ) = $$

$$ = \ \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty f( x _ {1} - u _ {1} \dots x _ {n} - u _ {n} ) g( u _ {1} \dots u _ {n} ) \times $$

$$ \times du _ {1} \dots du _ {n\ } = $$

$$ = \ \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty f( u _ {1} \dots u _ {n} ) g( x _ {1} - u _ {1} \dots x _ {n} - u _ {n} ) \times $$

$$ \times \ du _ {1} \dots du _ {n} . $$

Let $ X = ( X _ {1} \dots X _ {n} ) $ and $ Y = ( Y _ {1} \dots Y _ {m} ) $ be random vectors taking values in $ \mathbf R ^ {n} $ and $ \mathbf R ^ {m} $ $ ( n, m \geq 1) $ and having probability densities $ f $ and $ g $ respectively, and let $ Z = ( X _ {1} \dots X _ {n} , Y _ {1} \dots Y _ {m} ) $ be a random vector in $ \mathbf R ^ {n+} m $. If then $ X $ and $ Y $ are independent, $ Z $ has the probability density $ h $, which is called the joint probability density of the random vectors $ X $ and $ Y $, where

$$ \tag{1 } h( t _ {1} \dots t _ {n+} m ) = f( t _ {1} \dots t _ {n} ) g( t _ {n+} 1 \dots t _ {n+} m ). $$

Conversely, if $ Z $ has a probability density that satisfies (1), then $ X $ and $ Y $ are independent.

The characteristic function $ \phi $ of a random vector $ X $ having a probability density $ f $ is expressed by

$$ \phi ( t _ {1} \dots t _ {n} ) = $$

$$ = \ \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty e ^ {i( t _ {1} x _ {1} + \dots + t _ {n} x _ {n} ) } f( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n} , $$

where if $ \phi $ is absolutely integrable then $ f $ is a bounded continuous function, and

$$ f( x _ {1} \dots x _ {n} ) = $$

$$ = \ \frac{1}{( 2 \pi ) ^ {n} } \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty e ^ {- i( t _ {1} x _ {1} + \dots + t _ {n} x _ {n} ) } \phi ( t _ {1} \dots t _ {n} ) dt _ {1} \dots dt _ {n} . $$

The probability density $ f $ and the corresponding characteristic function $ \phi $ are related also by the following relation (Plancherel's identity): The function $ f ^ { 2 } $ is integrable if and only if the function $ | \phi | ^ {2} $ is integrable, and in that case

$$ \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty f ^ { 2 } ( x _ {1} \dots x _ {n} ) dx _ {1} \dots dx _ {n\ } = $$

$$ = \ \frac{1}{( 2 \pi ) ^ {n} } \int\limits _ {- \infty } ^ \infty \dots \int\limits _ {- \infty } ^ \infty | \phi ( t _ {1} \dots t _ {n} ) | ^ {2} dt _ {1} \dots dt _ {n} . $$

Let $ ( \Omega , \mathfrak A) $ be a measurable space, and let $ \nu $ and $ \mu $ be $ \sigma $- finite measures on $ ( \Omega , \mathfrak A) $ with $ \nu $ absolutely continuous with respect to $ \mu $, i.e. $ \mu ( A) = 0 $ implies $ \nu ( A) = 0 $, $ A \in \mathfrak A $. In that case there exists on $ ( \Omega , \mathfrak A) $ a non-negative measurable function $ f $ such that

$$ \nu ( A) = \int\limits _ { A } f d \mu $$

for any $ A \in \mathfrak A $. The function $ f $ is called the Radon–Nikodým derivative of $ \nu $ with respect to $ \mu $, while if $ \nu $ is a probability measure, it is also the probability density of $ \nu $ relative to $ \mu $.

A concept closely related to the probability density is that of a dominated family of distributions. A family of probability distributions $ \mathfrak P $ on a measurable space $ ( \Omega , \mathfrak A) $ is called dominated if there exists a $ \sigma $- finite measure $ \mu $ on $ ( \Omega , \mathfrak A) $ such that each probability measure from $ \mathfrak P $ has a probability density relative to $ \mu $( or, what is the same, if each measure from $ \mathfrak P $ is absolutely continuous with respect to $ \mu $). The assumption of dominance is important in certain theorems in mathematical statistics.

References

[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)
[2] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)
[3] E.L. Lehmann, "Testing statistical hypotheses", Wiley (1986)
How to Cite This Entry:
Density of a probability distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_of_a_probability_distribution&oldid=46629
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article