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silicon2006@hotmail.com science forum beginner
Joined: 19 May 2006
Posts: 6

Posted: Thu Jun 15, 2006 8:37 am Post subject:
a question related to mutivariable normal distribution



(1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?
(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)? 

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john2 science forum beginner
Joined: 01 Feb 2006
Posts: 15

Posted: Thu Jun 15, 2006 1:08 pm Post subject:
Re: a question related to mutivariable normal distribution



silicon2006@hotmail.com wrote:
Quote:  (1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?
(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)?

A rank statistics problem [qv]. If x1..xn are independent, the CDF of
the largest sample is the nth power of the CDF of a single sample and
the PDF is not normal, though it might approximate it.
john2 

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Debashis Dash science forum beginner
Joined: 16 Jun 2006
Posts: 1

Posted: Fri Jun 16, 2006 12:00 am Post subject:
Re: a question related to mutivariable normal distribution



It is true that the distribution is not normal (gaussian to be more
general), but you can easilly find its CDF and PDF:
(capital F is the CDF and f is the pdf and subscript denotes the variable)
Let Y=max(X_1, X_2, ..., X_N)
Then, F_Y(a)=Pr(Y<=a)
=Pr(max(X_1, X_2, ..., X_N)<=a)
=Pr(X_1<=a).Pr(X_2<=a)...Pr(X_N<=a)
={F_X(a)}^N
Hence, f_Y(a)=d/da(F_Y(a))=N(F_X(a))^(N1).f_X(a)
Which is not a surprise.
DD
"john2" <john2@8889.fsnet.co.uk> wrote in message
news:e6rm4n$c7g$1@newsg4.svr.pol.co.uk...
Quote:  silicon2006@hotmail.com wrote:
(1) If x1, x2, x3, ..., xn all follow the same normal distribution (a,
sigma),
does x=maximum(x1, x2, ..., xn) also follow normal distribution?
(2) If x=maximum(x1, x2, ..., xn) does follow normal distribution,
what is mean(x) and std(x)?
A rank statistics problem [qv]. If x1..xn are independent, the CDF of the
largest sample is the nth power of the CDF of a single sample and the PDF
is not normal, though it might approximate it.
john2 


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Ray Koopman science forum Guru Wannabe
Joined: 25 Mar 2005
Posts: 216

Posted: Fri Jun 16, 2006 7:48 am Post subject:
Re: a question related to mutivariable normal distribution



Debashis Dash wrote:
Quote:  ,,,the distribution is not normal (gaussian to be more general)...

How do you see "gaussian" as more general than "normal"? 

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Google


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