30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS
30.9.4 The chi-squared distributionIn subsection 30.6.2, we showed that ifXis Gaussian distributed with meanμand
varianceσ^2 , such thatX∼N(μ, σ^2 ), then the random variableY=(x−μ)^2 /σ^2
is distributed as the gamma distributionY ∼γ(^12 ,^12 ). Let us now considern
independent Gaussian random variablesXi∼N(μi,σi^2 ),i=1, 2 ,...,n, and define
the new variable
χ^2 n=∑ni=1(Xi−μi)^2
σ^2 i. (30.122)
Using the result (30.121) for multiple gamma distributions,χ^2 nmust be distributed
as the gamma variateχ^2 n∼γ(^12 ,^12 n), which from (30.118) has the PDF
f(χ^2 n)=1
2
Γ(^12 n)(^12 χ^2 n)(n/2)−^1 exp(−^12 χ^2 n)=1
2 n/^2 Γ(^12 n)(χ^2 n)(n/2)−^1 exp(−^12 χ^2 n). (30.123)This is known as thechi-squared distributionof ordernand has numerous
applications in statistics (see chapter 31). Settingλ=^12 andr=^12 nin (30.120),
we find that
E[χ^2 n]=n, V[χ^2 n]=2n.An important generalisation occurs when thenGaussian variablesXiarenotlinearly independent but are instead required to satisfy a linear constraint of the
form
c 1 X 1 +c 2 X 2 +···+cnXn=0, (30.124)in which the constantsciare not all zero. In this case, it may be shown (see
exercise 30.40) that the variableχ^2 ndefined in (30.122) is still described by a chi-
squared distribution, but one of ordern−1. Indeed, this result may be trivially
extended to show that if thenGaussian variablesXisatisfymlinear constraints
of the form (30.124) then the variableχ^2 ndefined in (30.122) is described by a
chi-squared distribution of ordern−m.
30.9.5 The Cauchy and Breit–Wigner distributionsA random variableX(in the range−∞to∞) that obeys theCauchy distribution
is described by the PDF
f(x)=1
π1
1+x^2.