### Appendix V. Cumulative Gaussian and Chi-Squared
Distributions

The ^{2} confidence
limit is the probability of Chi-squared exceeding the observed value; i.e.,

where *P*_{p} for *p* degrees of freedom is given by
Eq. (30a).

__Gaussian Confidence Limits__

Let ^{2} =
[*x* / ]^{2}.
Then for *n*_{D} = 1,

Thus CL for *n*_{D} is twice the area under a single
Gaussian tail. For example the *n*_{D} = 1 curve for
^{2} = 4 has a value of
*CL* = 0.046.
This means that the probability of getting
| *x*|
2 is 4.6% for
a Gaussian distribution.