16.5 - The Standard Normal and The Chi | Normal distribution to chi-square distribution
Wehaveonemoretheoreticaltopictoaddressbeforegettingbacktosomepracticalapplicationsonthenextpage,andthatistherelationshipbetweenthenormaldistributionandthechi-squaredistribution.Thefollowingtheoremclarifiestherelationship.If(X)isnormallydistributedwithmean(mu)andvariance(sigma2>0),then:(V=left(dfrac{X-mu}{sigma}ight)2=Z2)isdistributedasachi-squarerandomvariablewith1degreeoffreedom.ProofToprovethistheorem,weneedtoshowthatthep.d.f.oftherandomvariable(V)isthesameasthep.d.f.ofachi-squarerandom...
We have one more theoretical topic to address before getting back to some practical applications on the next page, and that is the relationship between the normal distribution and the chi-square distribution. The following theorem clarifies the relationship.
If (X) is normally distributed with mean (mu) and variance (sigma2>0), then:
(V=left(dfrac{X-mu}{sigma} ight)2=Z2)
is distributed as a chi-square random variable with 1 degree of freedom.
ProofTo prove this theorem, we need to show that the p.d.f. of the random variable (V) is the same as the p.d.f. of a chi-square random variable with 1 degree of freedom. That is, we need to show that:
(g(v)=dfrac{1}{Gamma(1/2)2{1/2}}v{frac{1}{2}-1} e{-v/2})
The strategy well take is to find (G(v)), the cumulative distribution function of (V), and then differentiate it to get (g(v)), the probability density functi...