25.3 - Sums of Chi | Sum of chi square distribution
Let(Z_1,Z_2,ldots,Z_n)havestandardnormaldistributions,(N(0,1)).Iftheserandomvariablesareindependent,then:(W=Z2_1+Z2_2+cdots+Z2_n)followsa(chi2(n))distribution.ProofRecallthatif(Z_isimN(0,1)),then(Z_i2simchi2(1))for(i=1,2,ldots,n).Then,bytheadditivepropertyofindependentchi-squares:(W=Z2_1+Z2_2+cdots+Z2_nsimchi2(1+1+cdots+1)=chi2(n))Thatis,(Wsimchi2(n)),aswastobeproved.
Let (Z_1, Z_2, ldots, Z_n) have standard normal distributions, (N(0,1)). If these random variables are independent, then:
(W=Z2_1+Z2_2+cdots+Z2_n)
follows a (chi2(n)) distribution.
ProofRecall that if (Z_isim N(0,1)), then (Z_i2sim chi2(1)) for (i=1, 2, ldots, n). Then, by the additive property of independent chi-squares:
(W=Z2_1+Z2_2+cdots+Z2_n sim chi2(1+1+cdots+1)=chi2(n))
That is, (Wsim chi2(n)), as was to be proved.