These notes summarise the theory for the Maths Stats 245 module at Stellenbosch University. The notes include theory, worked examples and detailed explanations.
Wewilloftenorderobservedrandomvariables accordingto their magnitudesTheresulting
ordered variables are called orderstatistics
het4,42 Yn denote independent continues random variables with distributionfunctionFly
and density function We denotetheordered random variables by YenYea Yen
fly
where Yale4141 Yen
Ya min Yi Yn and Yen Max Yi Yn
Eg U Max Xi XzXoXx Xs AS
Xi 0.43 0.67 0.32 0.48 0.78 4 0,78
themaximum and minimum will havedistributions we want to findtheirdistributions
so that we can conduct inference on them
NBnotation Fu u CDFof U mustusethisnotation
fu u PDF of u mustusethisnotation
Forthefollowingsections we will need to know all methods of transformations
het U Max Yi Yn We want to findthe distributionof u
We View u as a transformation ontheXivariables and we usethe method of distribution
functions to find the distribution of U
ifthemaximum islessthan a particular
Value Uthenall Yi'smustbeless
Feiause
Wewantto find Falu U Max Yi yn u isthemaximumvalu
CDFthenwe will 4 P Usa the event Yinka will only
takederivative P Max y yay y occur if andonly ifthe
find PDF P Ysu 4214 Yn events Yifu occurforevery
P Yiu PYau PYnEU y for all I 1,2 n
Yiareindependent Fu u Plusa P Yiu PYasu Fyu
and Petitul FyYea P Ysu
Nowtake derivative of Falu to
get PDF
knowproof
fulu dduFcu
D flu Fycal
Nowwelookat the minimum
Let V min Yi Yn For Viv thiswillonlyoccur ifandonly if all Yi's
are YilV
Therefore I Fu v P Vsv
p min Yi Yn v
, P YiuYasu In v all Yimust be largerthan U
AllYiareindup PLY v PYalu PCYniv
stakeproduct I PCYicu LI Peyser Il PCYner
I Fyly
pyou
I Fufu I Fyly now we differentiate
Frfr I I Fyly
frfr n l fyly Cfyld
N I fyly Fy v
PDFandCDFformaximum andminimum
maximum fulu D flu Fycal and Fula Fylul
minimum s fu v n l fyly fych and Fulu i Fylul
Example X is a randomvariable and X unit ai know PDF'sof
het U Max Xi Xn We want fucul distributions
Willnot formula
geta
We know fufu R.fylul.Fyc.at sheet
054
fx x I Otoe
otherwise
x
L o
I
occo
211
, fu u n i U Of us I
plug uinto Fx x youget a
pluguinto
fx x you and formula is fx x
simplyget1
u betacan
un
fisition be Ualso of us
fucalyn g otherwise
T atB 1 Bl we have an Bel
ya i
g
TIATIB
I na
ago
Pinta y
1
N i un
Alternative distributionforthemaximum 1
way of proving
Wefollowthe differential technique Wethinkabouttheprobability density function of a
continues variable x at a specific point to be equal proportionally to the probability
of the Variable value closetothepoint
In otherwords PlaceXi actdad fixdx
Therefore
to find flu we observe that Uf u s utdu and that happens if one ofthe
N Xi'sfalls withinthe interval u Utdu andtherest oftheXi's are totheleft of u
U isthemaximum if u falls inthe intervalthen all other Xi'sare lessthan u
and we get
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