The numerical value of the likelihood function at
(*) can, in principle, be used as
a check on whether one is using the correct type of function for
*f* (; *x*). If one is
using the wrong f, the likelihood function will be lower in
height and of greater width. In principle, one can calculate,
using direct probability, the distribution of
(*) assuming a particular true
*f* (_{0},
*x*). Then the probability of getting an
(*) smaller than the value
observed would be a useful
indication of whether the wrong type of function for f had been
used. If for a particular experiment one got the answer that
there was one chance in 10^{4} of getting such a low value of
(*), one would seriously question
either the experiment or the function
*f* (;*x*) that
was used.

In practice, the determination of the distribution of
(*) is usually an impossibly
difficult numerical integration
in *N*-dimensional space. However, in the special case of the
least-square problem, the integration limits turn out to be
the radius vector in p-dimensional space. In this case we use
the distribution of
*S*(*) rather than of
(*). We shall first consider the
distribution of
*S*(_{0}).
According to Eqs. (23) and (24) the probability element is

Note that *S* =
^{2},
where is the
magnitude of the radius vector
in *p*-dimensional space. The volume of a *p*-dimensional sphere
is *U*
_{p}. The
volume element in this space is then

Thus

The normalization is obtained by integrating from *S* = 0 to
*S* = .

(30a) |

where
*S*
*S*(_{0}).

This distribution is the well-known
^{2} distribution with
*p*
degrees of freedom. ^{2}
tables of

for several degrees of freedom are commonly available - see Appendix V for plots of the above integral.

From the definition of *S* (Eq. (24)) it is obvious that
_{0} = *p*. One
can show, using Eq. (29) that
= 2*p*. Hence,
one should be suspicious if his experimental result gives an
*S*-value much greater than

Usually is not known. In such a case one is interested in the distribution of

Fortunately, this distribution is also quite simple. It is
merely the ^{2}
distribution of (*p* - *M*) degrees of freedom, where
*p* is the number of experimental points, and *M* is the number
of parameters solved for. Thus we haved

(31) |

Since the derivation of Eq. (31) is somewhat lengthy, it is given in Appendix II.

__Example 8__

Determine the ^{2}
probability of the solution to Example 6.

According to the ^{2}
table for one degree of freedom the probability of getting
*S** > 0.674 is 0.41. Thus the experimental data
are quite consistent with the assumed theoretical shape of

__Example 9 Combining Experiments__

Two different laboratories have measured the lifetime of
the *K*_{1}^{0} to be
(1.00 ± 0.01) × 10^{-10} sec and
(1.04 ± 0.02) × 10^{10} sec
respectively. Are these results really inconsistent?

According to Eq. (6) the weighted mean is
* = 1.008 ×
10^{-10} sec.
(This is also the least squares solution for
_{KO}.

Thus

According to the ^{2}
table for one degree of freedom, the probability of getting
*S** > 3.2 is 0.074. Therefore, according
to statistics, two measurements of the same quantity should be
at least this far apart 7.4% of the time.