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The [[Probability distribution|probability distribution]] with probability density
 
The [[Probability distribution|probability distribution]] with probability density
  
<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/m/m063/m063130/m0631301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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p ( x)  = \
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\left \{
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\begin{array}{ll}
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{}  &{}  \\
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\sqrt {
 +
\frac{2} \pi
 +
}
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\frac{x  ^ {2} }{\sigma  ^ {3} }
 +
 
 +
e ^ {- x  ^ {2} / 2 \sigma  ^ {2} } ,  & x \geq  0,  \\
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0 ,  & x < 0 ,  \\
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\end{array}
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\right .
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$$
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depending on a parameter  $  \sigma > 0 $.  
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The [[Distribution function|distribution function]] of the Maxwell distribution has the form
  
depending on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631302.png" />. The [[Distribution function|distribution function]] of the Maxwell distribution has the form
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$$
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F ( x)  = \
 +
\left \{
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\begin{array}{ll}
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{}  &{}  \\
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2 \Phi \left (
 +
\frac{x} \sigma
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\right ) -
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\sqrt {
 +
\frac{2} \pi
 +
} {
 +
\frac{x} \sigma
 +
} e ^
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{- x  ^ {2} / 2 \sigma  ^ {2} } - 1 ,  & x \geq  0 ,  \\
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0 ,  & x < 0 ,  \\
 +
\end{array}
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\right.
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$$
  
<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/m/m063/m063130/m0631303.png" /></td> </tr></table>
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where  $  \Phi ( x) $
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is the standard [[Normal distribution|normal distribution]] function. The Maxwell distribution has positive coefficient of skewness; it is unimodal, the unique mode occurring at  $  x = \sqrt 2 \sigma $.
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The Maxwell distribution has finite moments of all orders; the mathematical expectation and variance are equal to  $  2 \sigma \sqrt {2 / \pi } $
 +
and  $  ( 3 \pi - 8 ) \sigma  ^ {2} / \pi $,
 +
respectively.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631304.png" /> is the standard [[Normal distribution|normal distribution]] function. The Maxwell distribution has positive coefficient of skewness; it is unimodal, the unique mode occurring at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631305.png" />. The Maxwell distribution has finite moments of all orders; the mathematical expectation and variance are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631307.png" />, respectively.
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If $X_{1}$, $X_{2}$ and $X_{3}$ are independent random variables having the normal distribution with parameters $0$
 +
and $\sigma^{2}$,
 +
then the random variable  $  \sqrt {X _ {1}  ^ {2} + X _ {2}  ^ {2} + X _ {3}  ^ {2} } $
 +
has a Maxwell distribution with density (*). In other words, a Maxwell distribution can be obtained as the distribution of the length of a random vector whose Cartesian coordinates in three-dimensional space are independent and normally distributed with parameters  $  0 $
 +
and $  \sigma  ^ {2} $.  
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631308.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m0631309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313010.png" /> are independent random variables having the normal distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313012.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313013.png" /> has a Maxwell distribution with density (*). In other words, a Maxwell distribution can be obtained as the distribution of the length of a random vector whose Cartesian coordinates in three-dimensional space are independent and normally distributed with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313015.png" />. The Maxwell distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313016.png" /> coincides with the distribution of the square root of a variable having the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063130/m06313017.png" />-distribution with three degrees of freedom (see also [[Rayleigh distribution|Rayleigh distribution]]). The Maxwell distribution is widely known as the velocity distribution of particles in statistical mechanics and physics. The distribution was first defined by J.C. Maxwell (1859) as the solution of the problem on the distribution of velocities of molecules in an ideal gas.
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The Maxwell distribution with $  \sigma = 1 $
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coincides with the distribution of the square root of a variable having the $  \chi  ^ {2} $-
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distribution with three degrees of freedom (see also [[Rayleigh distribution|Rayleigh distribution]]). The Maxwell distribution is widely known as the velocity distribution of particles in statistical mechanics and physics. The distribution was first defined by J.C. Maxwell (1859) as the solution of the problem on the distribution of velocities of molecules in an ideal gas.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</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></table>
+
<table>
 +
<TR><TD valign="top">[1]</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>
 +
</table>

Latest revision as of 11:10, 9 April 2023


The probability distribution with probability density

$$ \tag{* } p ( x) = \ \left \{ \begin{array}{ll} {} &{} \\ \sqrt { \frac{2} \pi } \frac{x ^ {2} }{\sigma ^ {3} } e ^ {- x ^ {2} / 2 \sigma ^ {2} } , & x \geq 0, \\ 0 , & x < 0 , \\ \end{array} \right . $$

depending on a parameter $ \sigma > 0 $. The distribution function of the Maxwell distribution has the form

$$ F ( x) = \ \left \{ \begin{array}{ll} {} &{} \\ 2 \Phi \left ( \frac{x} \sigma \right ) - \sqrt { \frac{2} \pi } { \frac{x} \sigma } e ^ {- x ^ {2} / 2 \sigma ^ {2} } - 1 , & x \geq 0 , \\ 0 , & x < 0 , \\ \end{array} \right. $$

where $ \Phi ( x) $ is the standard normal distribution function. The Maxwell distribution has positive coefficient of skewness; it is unimodal, the unique mode occurring at $ x = \sqrt 2 \sigma $. The Maxwell distribution has finite moments of all orders; the mathematical expectation and variance are equal to $ 2 \sigma \sqrt {2 / \pi } $ and $ ( 3 \pi - 8 ) \sigma ^ {2} / \pi $, respectively.

If $X_{1}$, $X_{2}$ and $X_{3}$ are independent random variables having the normal distribution with parameters $0$ and $\sigma^{2}$, then the random variable $ \sqrt {X _ {1} ^ {2} + X _ {2} ^ {2} + X _ {3} ^ {2} } $ has a Maxwell distribution with density (*). In other words, a Maxwell distribution can be obtained as the distribution of the length of a random vector whose Cartesian coordinates in three-dimensional space are independent and normally distributed with parameters $ 0 $ and $ \sigma ^ {2} $.

The Maxwell distribution with $ \sigma = 1 $ coincides with the distribution of the square root of a variable having the $ \chi ^ {2} $- distribution with three degrees of freedom (see also Rayleigh distribution). The Maxwell distribution is widely known as the velocity distribution of particles in statistical mechanics and physics. The distribution was first defined by J.C. Maxwell (1859) as the solution of the problem on the distribution of velocities of molecules in an ideal gas.

References

[1] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971)
How to Cite This Entry:
Maxwell distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maxwell_distribution&oldid=25970
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article