Physics and Engineering of Radiation Detection

9.4. Confidence Intervals 547

If the functionL(x) is normalized, then the denominator becomes 1 and the
probability is simply given by

P=

∫x 2

x 1

L(x)dx. (9.4.2)

This probability is actually the area under the curve ofL(x) versusxbetween
the pointsx 1 andx 2 and therefore depends on the choice of the confidence interval
(see Fig.9.4.1).

x 1 x 2 x

L(x)

Figure 9.4.1: The probability that a

valuexlies within a confidence interval of

(x 1 ,x 2 ) is obtained by dividing the area

under the curve (shaded section) by the

total area. If the distribution is nor-

malized then the denominator will be 1

and the shaded area will simply be the

required probability. This probability,

therefore, depends on the choice of con-

fidence interval. In practice, the proba-

bility is first selected and then the confi-

dence interval is obtained from the cumu-

lative distribution function of the proba-

bility distribution.

We saw above that the choice of confidence interval is arbitrary while the prob-
ability depends on it. Therefore one would assume that the interval is first chosen
and then the probability is calculated. However the general practice is quite the
opposite. If it is known that the process under consideration has a certain probabil-
ity distribution then the probability is first chosen and then the confidence interval
is deduced from some available tables or curves. For example, for Gaussian dis-
tribution, which is the most commonly used distribution, the tables of probability
integrals are used to find the confidence intervals.
Let us now take a look at the example of a normally distributed variablex
having meanμand varianceσ^2. We are interested in finding the probability that
the measured value lies betweenμ−δxandμ+δx. This probability, according to
the definition above, can be evaluated from

P =

1

σ

√

2 π

∫μ+δx

μ−δx

e−(x−μ)

(^2) / 2 σ 2
dx
= erf

(

δx

σ

√

2

)

. (9.4.3)

Hereerf(u)istheerror functionofu, whose values are available in tabulated
form in standard texts. To get a feeling of what different values ofP would mean