a question related to muti-variable normal distribution

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

(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)?

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

silicon2006@hotmail.com wrote:

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

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))^(N-1).f_X(a)

Which is not a surprise.

DD

"john2" <john2@8889.fsnet.co.uk> wrote in message
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Ray Koopman · science forum Guru Wannabe Joined: 25 Mar 2005 Posts: 216

Debashis Dash wrote:

Google