Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

0

0

0. 2

0. 4

0. 6

0. 8

− 4 − 224

x

f(x)

x 0 =0,

x 0 =0,

x 0 =2,

Γ=1 Γ=1

Γ=3

Figure 30.17 The PDFf(x) for the Breit–Wigner distribution for different

values of the parametersx 0 and Γ.

This is a special case of theBreit–Wigner distribution

f(x)=

1

π

1

2 Γ

1

4 Γ

(^2) +(x−x 0 ) 2 ,
which is encountered in the study of nuclear and particle physics. In figure 30.17,
we plot some examples of the Breit–Wigner distribution for several values of the
parametersx 0 and Γ.
We see from the figure that the peak (or mode) of the distribution occurs
atx=x 0. It is also straightforward to show that the parameter Γ is equal to
the width of the peak at half the maximum height. Although the Breit–Wigner
distribution is symmetric about its peak, it does not formally possess a mean since
the integrals
∫ 0
−∞xf(x)dxand
∫∞
0 xf(x)dxboth diverge. Similar divergences occur
for all higher moments of the distribution.
30.9.6 The uniform distribution
Finally we mention the very simple, but common,uniform distribution,which
describes a continuous random variable that has a constant PDF over its allowed
range of values. If the limits onXareaandbthen
f(x)=
{
1 /(b−a)fora≤x≤b,
0otherwise.
The MGF of the uniform distribution is found to be
M(t)=
ebt−eat
(b−a)t
,