Editing Tolerance interval (section)

== Definition ==
{{expert needed|Statistics|reason=Definition needs to be contrasted and discussed against definition of a [[prediction interval]]|date=May 2024}}

Assume observations or [[random variate]]s <math>\mathbf{x}=(x_1,\ldots,x_n)</math> as realization of independent random variables <math>\mathbf{X}=(X_1,\ldots,X_n)</math> which have a common distribution <math>F_\theta</math>, with unknown parameter <math>\theta</math>.
Then, a tolerance interval with endpoints <math>(L(\mathbf{x}), U(\mathbf{x})]</math> which has the defining property:<ref name="Meeker&Hahn">{{cite book | last1=Meeker | first1=W.Q. | last2=Hahn | first2=G.J. | last3=Escobar | first3=L.A. | title=Statistical Intervals: A Guide for Practitioners and Researchers | publisher=Wiley | series=Wiley Series in Probability and Statistics | year=2017 | isbn=978-0-471-68717-7 | url=https://books.google.com/books?id=y3o0DgAAQBAJ | access-date=2024-11-05 | page=}}</ref>

:<math>\inf_\theta\{{\Pr}_\theta\left(F_\theta(U(\mathbf{X})) - F_\theta(L(\mathbf{X})\right) \ge p)\} = 100(1-\alpha)</math>
where <math>\inf\{\}</math> denotes the [[infimum]] function. 

This is in contrast to a prediction interval with endpoints <math>[l(\mathbf{x}), u(\mathbf{x})]</math> which has the defining property:<ref name="Meeker&Hahn"/>
:<math> \inf_\theta\{{\Pr}_\theta(X_0 \in [l(\mathbf{X}), u(\mathbf{X})])\}= 100(1-\alpha)</math>.
Here, <math>X_0</math> is a random variable from the same distribution <math>F_\theta</math> but independent of the first <math>n</math> variables.

Notice <math>X_0</math> is {{em|not}} involved in the definition of tolerance interval, which deals only with the first sample, of size ''n''.