Here we are concerned with the case in which an event must be one of two classes, such as up or down, forward or back, positive or negative, etc. Let p be the probability for an event of Class 1. Then (1 - p) is the probability for Class 2, and the joint probability for observing N1 events in Class 1 out of N total events is
| The binomial distribution | (14) |
Note that
j=1N
p(j, N) = [p + (1 - p)]N =
1. The factorials correct
for the fact that we are not interested in the order in which
the events occurred. For a given experimental result of
N1 out of N events in Class 1, the likelihood
function
(p) is then
| (15) |
| (16) |
From Eq. (15) we have
| (17) |
From (16) and (17):
| (18) |
The results, Eqs. (17) and (18), also happen to be the same as those using direct probability. Then
|
and
|
Example 4
In Example 1 on the µ-e decay angular distribution we found that
|
is the error on the asymmetry parameter
. Suppose that the
individual cosine, xi, of each event is not known. In this
problem all we know is the number up vs. the number down. What
then is
? Let p be the probability
of a decay in the up hemisphere; then we have
|
By Eq. (18),
|
For small this is
=
sqrt[4 / N] as compared to
sqrt[3 / N] when the full information is used.