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| + | $#C+1 = 91 : ~/encyclopedia/old_files/data/D031/D.0301110 Density of a probability distribution, |
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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> |
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) |