Physics and Engineering of Radiation Detection

9.3. Probability 531

then the expression for the expectation value becomes

E(g(x)) =

∫∞

−∞

g(x)f(x)dx. (9.3.6)

In this book, unless otherwise stated, we will assume that the p.d.f has already

been normalized.

Moments:Using the definition of the expectation value given above, we can

compute the the expectation value ofxn. This quantity is called thenthmo-

ment ofxand is defined as

αn=

∫∞

−∞

xnf(x)dx. (9.3.7)

The most commonly used moment is the first moment (generally represented

byμ), which simply represents the weighted mean ofx. For a normalized

distribution functionf(x), it is given by

μ≡α 1 =

∫∞

−∞

xf(x)dx. (9.3.8)

α 1 is also called the expectation value ofx(generally represented by E(x)), as

this is the value that we should expect if we perform another measurement.

How confident we are about thisexpectation, depends on the distribution func-

tion.

Central Moments:Thenthcentral moment of any variablexabout its mean

μis defined as the expectation value of (x−μ)n.Ifxfollows a probability

distribution functionf(x), itsnthcentral moment can be calculated from

mn=

∫∞

−∞

(x−μ)nf(x)dx. (9.3.9)

The varianceσ≡m 2 is the most commonly used central moment. According

to the above definition, it is given by

σ=

∫∞

−∞

(x−μ)^2 f(x)dx. (9.3.10)

Variance quantifies the spread of the values around their mean and is used to

characterize the level of uncertainty in a measurement.

It can be shown that

σ^2 = E(x^2 )−(E(x))^2

= α 2 −μ^2. (9.3.11)

Characteristic Function:Up until now we have assumed that the distribu-

tion function can be explicitly written and easily manipulated to determine

the moments. Unfortunately this is not always the case. Sometimes we come