Mathematical Methods for Physics and Engineering - Matematica.NET

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS30.9.4 The chi-squared distributionIn subsection 30.6.2, we showed that if X is Gaussian distributed with mean µ andvariance σ 2 , such that X ∼ N(µ, σ 2 ), then the random variable Y =(x − µ) 2 /σ 2is distributed as the gamma distribution Y ∼ γ( 1 2 , 1 2). Let us now consider nindependent Gaussian random variables X i ∼ N(µ i ,σi 2 ), i =1, 2,...,n, and definethe new variableχ 2 n =n∑ (X i − µ i ) 2. (30.122)i=1Using the result (30.121) for multiple gamma distributions, χ 2 n must be distributedas the gamma variate χ 2 n ∼ γ( 1 2 , 1 2n), which from (30.118) has the PDF1f(χ 2 2n)=Γ( 1 2 n)(1 2 χ2 n) (n/2)−1 exp(− 1 2 χ2 n)1=2 n/2 Γ( 1 n) (n/2)−1 exp(− 12 n)(χ2 2 χ2 n). (30.123)This is known as the chi-squared distribution of order n and has numerousapplications in statistics (see chapter 31). Setting λ = 1 2 and r = 1 2n in (30.120),we find thatE[χ 2 n]=n,σ 2 iV [χ 2 n]=2n.An important generalisation occurs when the n Gaussian variables X i are notlinearly independent but are instead required to satisfy a linear constraint of theformc 1 X 1 + c 2 X 2 + ···+ c n X n =0, (30.124)in which the constants c i are not all zero. In this case, it may be shown (seeexercise 30.40) that the variable χ 2 n defined in (30.122) is still described by a chisquareddistribution, but one of order n − 1. Indeed, this result may be triviallyextended to show that if the n Gaussian variables X i satisfy m linear constraintsof the form (30.124) then the variable χ 2 n defined in (30.122) is described by achi-squared distribution of order n − m.30.9.5 The Cauchy and Breit–Wigner distributionsA random variable X (in the range −∞ to ∞) that obeys the Cauchy distributionis described by the PDFf(x) = 1 π119311+x 2 .