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A statistical procedure in which a decision is randomly taken. Suppose that, given a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773901.png" /> of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773902.png" /> with values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773904.png" />, one has to choose a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773905.png" /> from a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773906.png" />, and suppose that a family of so-called transition probability distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773908.png" />, has been defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r0773909.png" /> such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739010.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739011.png" />-measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739012.png" /> for every fixed event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739013.png" />. Then randomization is the statistical procedure of decision taking in which, given a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739015.png" />, the decision is made by drawing lots subject to the probability law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077390/r07739016.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Chentsov] Čentsov,   "Statistical decision rules and optimal inference" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR></table>



====Comments====
The statistical procedure of randomization is also called a randomized decision rule.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.O. Berger,   "Statistical decision theory and Bayesian analysis" , Springer  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>