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Random intervals, constructed for independent identically-distributed random variables with unknown distribution function  $  F ( x) $, 
containing with given probability  $  \gamma $
at least a proportion  $  p $ ($  0 < p < 1 $) 
of the probability measure  $  dF $.

Let  $  X _ {1} \dots X _ {n} $
be independent and identically-distributed random variables with unknown distribution function  $  F ( x) $, 
and let  $  T _ {1} = T _ {1} ( X _ {1} \dots X _ {n} ) $, 
$  T _ {2} = T _ {2} ( X _ {1} \dots X _ {n} ) $
be statistics such that, for a number  $  p $(
$  0 < p < 1 $) 
fixed in advance, the event  $  \{ F ( T _ {2} ) - F ( T _ {1} ) > p \} $
has a given probability  $  \gamma $, 
that is,

$$ \tag{1 }
{\mathsf P} \left \{
\int\limits _ { T _ {1} } ^ { {T _ 2 } }
dF ( x) \geq  p
\right \}  =  \gamma .
$$

In this case the random interval  $  ( T _ {1} , T _ {2} ) $
is called a  $  \gamma $-
tolerance interval for the distribution function  $  F ( x) $, 
its end points  $  T _ {1} $
and  $  T _ {2} $
are called tolerance bounds, and the probability  $  \gamma $
is called a confidence coefficient. It follows from (1) that the one-sided tolerance bounds  $  T _ {1} $
and  $  T _ {2} $(
i.e. with  $  T _ {2} = + \infty $, 
respectively  $  T _ {1} = - \infty $) 
are the usual one-sided confidence bounds with confidence coefficient  $  \gamma $
for the quantiles  $  x _ {1 - p }  = F ^ { - 1 } ( 1 - p) $
and  $  x _ {p} = F ^ { - 1 } ( p) $, 
respectively, that is,

$$ 
{\mathsf P} \{ x _ {1 - p }  \in [ T _ {1} , + \infty ) \}  =  \gamma ,
$$

$$ 
{\mathsf P} \{ x _ {p} \in (- \infty , T _ {2} ] \}  =  \gamma .
$$

Example. Let  $  X _ {1} \dots X _ {n} $
be independent random variables having a normal distribution  $  N ( a, \sigma  ^ {2} ) $
with unknown parameters  $  a $
and  $  \sigma  ^ {2} $. 
In this case it is natural to take the tolerance bounds  $  T _ {1} $
and  $  T _ {2} $
to be functions of the [[sufficient statistic]]  $  ( \overline{X}\; , S  ^ {2} ) $, 
where

$$ 
\overline{X}\;  = \ 
{
\frac{X _ {1} + \dots + X _ {n} }{n}
 } ,\ \ 
S  ^ {2}  = \ 
{
\frac{1}{n - 1 }
 }
\sum _ {i = 1 } ^ { n } 
( X _ {i} - \overline{X}\; )  ^ {2} .
$$

Specifically, one takes  $  T _ {1} = \overline{X}\; - kS  ^ {2} $
and  $  T _ {2} = \overline{X}\; + kS  ^ {2} $, 
where the constant  $  k $, 
called the tolerance multiplier, is obtained as the solution to the equation

$$ 
{\mathsf P} \left \{
\Phi \left (
{
\frac{\overline{X}\; + kS - a } \sigma 
 }
\right ) - \Phi \left (
{
\frac{\overline{X}\; - kS - a } \sigma 
 }
\right ) \geq  p
\right \}  =  \gamma ,
$$

where  $  \Phi ( x) $
is the distribution function of the standard normal law; moreover,  $  k = k ( n, \gamma , p) $
does not depend on the unknown parameters  $  a $
and  $  \sigma  ^ {2} $. 
The tolerance interval constructed in this way satisfies the following property: With confidence probability  $  \gamma $
the interval  $  ( \overline{X}\; - kS  ^ {2} , \overline{X}\; + kS  ^ {2} ) $
contains at least a proportion  $  p $
of the probability mass of the normal distribution of the variables  $  X _ {1} \dots X _ {n} $.

Assuming the existence of a probability density function  $  f ( x) = F ^ { \prime } ( x) $, 
the probability of the event  $  \{ F ( T _ {2} ) - F ( T _ {1} ) \geq  p \} $
is independent of  $  F ( x) $
if and only if  $  T _ {1} $
and  $  T _ {2} $
are order statistics (cf. [[Order statistic|Order statistic]]). Precisely this fact is the basis of a general method for constructing non-parametric, or distribution-free, tolerance intervals. Let  $  X  ^ {(*)} = ( X _ {( n1)} \dots X _ {( nn)} ) $
be the vector of order statistics constructed from the sample  $  X _ {1} \dots X _ {n} $
and let

$$ 
T _ {1}  =  X _ {( nr)} ,\ \ 
T _ {2}  =  X _ {( ns)} ,\ \ 
1 \leq  r < s \leq  n.
$$

Since the random variable  $  F ( X _ {( ns)} ) - F ( X _ {( nr)} ) $
has the beta-distribution with parameters  $  s - r $
and  $  n - s + r + 1 $, 
the probability of the event  $  \{ F ( X _ {( ns)} ) - F ( X _ {( nr)} ) \geq  p \} $
can be calculated as the integral  $  I _ {1 - p }  ( n - s + r + 1, s - r) $, 
where  $  I _ {x} ( a, b) $
is the incomplete beta-function, and hence in this case instead of (1) one obtains the relation

$$ \tag{2 }
I _ {1 - p }  ( n - s + r + 1, s - r)  =  \gamma ,
$$

which allows one, for given  $  \gamma $, 
$  p $
and  $  n $, 
to define numbers  $  r $
and  $  s $
so that the order statistics  $  X _ {( nr)} $
and  $  X _ {( ns)} $
are the tolerance bounds of the desired tolerance interval. Moreover, for given  $  \gamma $, 
$  p $, 
$  r $, 
relation (2) allows one to determine the size  $  n $
of the collection  $  X _ {1} \dots X _ {n} $
necessary for the relation (2) to hold. There are statistical tables available for solving such problems.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Wilks,   "Mathematical statistics" , Wiley  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. David,   "Order statistics" , Wiley  (1981)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.B. Murphy,   "Non-parametric tolerance limits"  ''Ann. Math. Stat.'' , '''19'''  (1948)  pp. 581–589</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.N. Somerville,   "Tables for obtaining non-parametric tolerance limits"  ''Ann. Math. Stat.'' , '''29'''  (1958)  pp. 599–601</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Scheffé,   J.W. Tukey,   "Non-parametric estimation I. Validation of order statistics"  ''Ann. Math. Stat.'' , '''16'''  (1945)  pp. 187–192</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.A.S. Fraser,   "Nonparametric methods in statistics" , Wiley  (1957)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Wald,   J. Wolfowitz,   "Tolerance limits for a normal distribution"  ''Ann. Math. Stat.'' , '''17'''  (1946)  pp. 208–215</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Robbins,   "On distribution-free tolerance limits in random sampling"  ''Ann. Math. Stat.'' , '''15'''  (1944)  pp. 214–216</TD></TR></table>