Physics and Engineering of Radiation Detection

530 Chapter 9. Essential Statistics for Data Analysis

be written as
P(x)=f(x;θ)dx.

Heref(x, θ) is called theprobability density functionor simply p.d.f. Some physicists
prefer to call itdistribution function. A p.d.f. may depend on several parameters,
which we have collectively represented byθ. Generallyθis unknown and its value
is determined through measurements ofx.The functionf(x;θ) need not be contin-
uous, though. It can take discrete values, in which case it will itself represent the
probability. The advantage of using a p.d.f. is that it enables us to predict a number
of variables related to the outcome of an experiment, such as the mean and the fre-
quency with which any random data will take on some particular value or lie within
a range of values.
Any p.d.f. must first be normalized before use. This can be done by noting that
the maximum probability of any event occurring is always 1. Hence we can integrate
the p.d.f. over all space to get the normalization constantN,thatis

N

∫∞

−∞

f(x, θ)dx=1.

C.1 QuantitiesDerivablefromaP.D.F

A probability density function is very convenient in terms of evaluating different
quantities related to its independent parameter.

Cumulative Distribution Function:Sometimes we are interested in finding

the probability of the occurrence of an event up to a certain value of the inde-

pendent parameter. This can be done by the so called cumulative distribution

functionF(a), which gives the probability that the variablexcan take any

valueuptoavaluea. It is defined by

F(a)≡F(−∞<x≤a)=

∫a

−∞

f(x)dx. (9.3.2)

If the functionf(x) is normalized then the probability thatxcan take any

value fromaonwards can be obtained from

F(a≤x<∞)=1−F(a). (9.3.3)

Expectation Value:The expectation value of any functiong(x) is obtained

by taking its weighted mean with the distribution function of its random vari-

able. For any general distribution functionf(x), it is given by

E(g(x)) =

∫∞

−∞∫ g(x)f(x)dx

∞

−∞f(x)dx

. (9.3.4)

If the functionf(x) has already been normalized, that is

∫∞

−∞

f(x)dx=1, (9.3.5)