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)