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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773401.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773402.png" /> into itself''
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A [[Random variable|random variable]] taking values in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773403.png" /> of all single-valued mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773404.png" /> into itself. The random mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773405.png" /> for which the probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773406.png" /> is positive only for one-to-one mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773407.png" /> are called random permutations of degree (order) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773408.png" />. The most thoroughly studied random mappings are those for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r0773409.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r07734010.png" />. A realization of such a random mapping is the result of a simple random selection from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077340/r07734011.png" />.
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''$\sigma$ of a set $X = \{1,2,\ldots,n\}$ into itself''
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A [[random variable]] taking values in the set $\Sigma_n$ of all single-valued mappings of $X$ into itself. The random mappings $\sigma$ for which the probability $\mathsf{P}\{\sigma=s\}$ is positive only for one-to-one mappings $s$ are called random permutations of degree (order) $n$. The most thoroughly studied random mappings are those for which $\mathsf{P}\{\sigma=s\} = n^{-n}$ for all $s \in\Sigma_n$. A realization of such a random mapping is the result of a simple random selection from $\Sigma_n$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kolchin,  "Random mappings" , Optim. Software  (1986)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  V.F. Kolchin,  "Random mappings" , Optim. Software  (1986)  (Translated from Russian)</TD></TR>
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</table>

Revision as of 21:29, 24 December 2015


$\sigma$ of a set $X = \{1,2,\ldots,n\}$ into itself

A random variable taking values in the set $\Sigma_n$ of all single-valued mappings of $X$ into itself. The random mappings $\sigma$ for which the probability $\mathsf{P}\{\sigma=s\}$ is positive only for one-to-one mappings $s$ are called random permutations of degree (order) $n$. The most thoroughly studied random mappings are those for which $\mathsf{P}\{\sigma=s\} = n^{-n}$ for all $s \in\Sigma_n$. A realization of such a random mapping is the result of a simple random selection from $\Sigma_n$.

References

[1] V.F. Kolchin, "Random mappings" , Optim. Software (1986) (Translated from Russian)
How to Cite This Entry:
Random mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Random_mapping&oldid=37097
This article was adapted from an original article by V.F. Kolchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article