When *M* parameters are to be determined from a single experiment
containing *N* events, the error formulas of the preceding
section are applicable only in the rare case in which the errors
are uncorrelated.. Errors are uncorrelated only for
= 0 for all
cases with
*i* *j*. For the
general case we Taylor-expand
*w*() about
(*):

where

and

(9) |

The second term of the expansion vanishes because
ð*w* / ð = 0
are the equations for *

Neglecting the higher-order terms, we have

(an M-dimensional Gaussian surface). As before, our error
formulas depend on the approximation that
() is
Gaussian-like in the region
_{i}
_{i}*. As mentioned in
Section 4, if the
statistics are so poor that this is a poor approximation, then
one should merely present a plot of
(). (see Appendix IV).

According to Eq. (9), __ H__ is a symmetric matrix. Let

(10) |

Let and . The element of probability in the -space is

Since
|__ U__| = 1 is the Jacobian relating the volume elements

Now that the general M-dimensional Gaussian surface has been put in the form of the product of independent one-dimensional Gaussians we have

Then

According to Eq. (10),
__ H__ =

Maximum Likelihood Errors, M parameters (11) |

(A rule for calculating the inverse matrix
*H*^{-1} is

If we use the alternate notation
__ V__ for the error matrix

(11a) |

__Example 2__

Assume that the ranges of monoenergetic particles are
Gaussian-distributed with mean range
_{1} and straggling
coefficient
_{2} (the standard
deviation). *N* particles having ranges
*x*_{1},..., *x*_{N} are observed. Find
_{1}*,
_{2}*, and their
errors . Then

The maximum-likelihood solution is obtained by setting the above two equations equal to zero.

The reader may remember a standard-deviation formula in which *N*
is replaced by (*N* - 1):

This is because in this case the most probable value,
_{2}*, and
the mean,
_{2} ,
do not occur at the same place. Mean
values of
such quantities are studied in Section 16.
The matrix H is
obtained by evaluating the following quantities at
_{1}* and
_{2}*:

According to Eq. (11), the errors on
_{1} and
_{2} are the
square
roots of the diagonal elements of the error matrix, *H*^{-1}:

(this is sometimes
called the error of the error). |

We note that the error of the mean is
1/sqrt[*N*]
where
=
_{2} is
the standard deviation. The error on the determination of
is
/sqrt[2*N*].

__Correlated Errors__

The matrix
*V*_{ij}
is
defined as the error matrix (also called the covariance matrix of
). In Eq. 11
we have shown that
__ V__ =

In the more common situation there will be one or more
off-diagonal elements to
__ H__ and the errors are correlated
(