15.8 - Chi | MGF of chi square
Chi-squareddistributionsareveryimportantdistributionsinthefieldofstatistics.Assuch,ifyougoontotakethesequelcourse,Stat415,youwillencounterthechi-squareddistributionsquiteregularly.Inthiscourse,wellfocusjustonintroducingthebasicsofthedistributionstoyou.InStat415,youllseeitsmanyapplications.Asitturnsout,thechi-squaredistributionisjustaspecialcaseofthegammadistribution!Letstakealook. Chi-squareDistributionwith(r)degreesoffreedomLet(X)followagammadistributionwith(heta=2)and(alpha=frac{r}{2}),whe...
Chi-squared distributions are very important distributions in the field of statistics. As such, if you go on to take the sequel course, Stat 415, you will encounter the chi-squared distributions quite regularly. In this course, well focus just on introducing the basics of the distributions to you. In Stat 415, youll see its many applications.
As it turns out, the chi-square distribution is just a special case of the gamma distribution! Lets take a look.
Chi-square Distribution with (r) degrees of freedomLet (X) follow a gamma distribution with ( heta=2) and (alpha=frac{r}{2}), where (r) is a positive integer. Then the probability density function of (X) is:
(f(x)=dfrac{1}{Gamma (r/2) 2{r/2}}x{r/2-1}e{-x/2})
for (x>0). We say that (X) follows a chi-square distribution with (r) degrees of freedom, denoted (chi2(r)) and read "chi-square-r."
There are, of course, an infinite n...