Physics and Engineering of Radiation Detection

9.3. Probability 535

tell us anything about how the other values ofhare spread out. This spread, also
known as the rms or root-mean-squared value, ofhabouth∗can be calculated from

h =

∣

∣

∣

∣

∫

(h−h∗)^2 Ldh

∫

Ldh

∣

∣

∣

∣

1 / 2

or

h =

[

−∂^2 l(h)

∂h^2

]− 1 / 2

. (9.3.22)

The second expression is extensively used to compute errors and due to its impor-
tance its derivation will be provided when we discuss the distribution functions later
in the chapter.
AlthoughL(h) can assume any shape but it can be shown that for large values of
N(that isN→∞), it approaches a Gaussian distribution as shown in Fig.9.3.2(b).
We will learn about this particular distribution in the next section when we take a
look at some of the commonly used distribution functions.

∆h

h*

h* h

h

L(h)

L(h)

(a)

(b)

N: very large

Figure 9.3.2: Typical likelihood func-

tions for relatively small (a) and very large

(b) number of data points. For largeN

the function approaches Gaussian distri-

bution.

A point worth noting here is that the case of low statistics needs careful attention.
The reason is that, as seen in Fig.9.3.2(b), the distribution can look fairly broad and
asymmetric if the available number of data points are low. In such a case, merely