Physics and Engineering of Radiation Detection

9.3. Probability 533

Median:In the beginning of this chapter we introducedmedianas a measure

of central tendency. Since we now know about probability density functions, we

are ready to appreciate the true meaning of this parameter. Median actually

represents the value at which the probability is 1/2, that is

P(xmed)=

∫xmed

−∞

f(x)dx=

∫∞

xmed

f(x)dx=

1

2

. (9.3.16)

Fig.9.3.1 represents this concept in graphical form. The computation of me-

dian becomes meaningful if the distribution is too skewed or has high negative

kurtosis, in which case the mean will not be a faithful representation of the

data.

xmed

xmed

(b)

(a)

x

x

f(x)

f(x)

Figure 9.3.1: Mode of a symmetric (a) and

a skewed (b) distribution. The mode is the

value ofxat which the probability is 1/2, that

is, the areas under the curve on the left and

right hand sides of the dotted line are equal.

C.2 MaximumLikelihoodMethod.................

The Bayesian approach we discussed earlier allows an experimenter to analyze the
data against some hypotheses. This can be done by the so calledmaximul likelihood
method. To elaborate on this methodology, we will look at a simplified example of
two hypotheses.
Let us suppose that we have two hypotheses about the outcome of an experiment.
With each of these hypotheses, we can associate a probability distribution function.