Density of a probability distribution

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)