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''probability density''

The derivative of the [[Distribution function|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|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====
<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>