Re: Probability distributions of maxima and minima
From: Jason Lenderman (jasonl_at_stat.ucla.edu)
Date: 03/29/05
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Date: Tue, 29 Mar 2005 00:16:29 -0800
Transform things (monotonically) so that you have n exp(1)-distributed
random variables by using appropriate CDF and/or inverse CDF functions. The
minimum of n exp(1)-distributed random variables has exp(1/n) distribution.
Now apply the inverse of the (previously used) transformation on a
exp(1/n)-distributed random variable to get the desired distribution. I
guess you should be able to get a fairly explicit formula for the CDF doing
this. In any case, the general idea seems to be that monotonic increasing
transformations commute with taking max/mins so you might as well put things
in a nice scale which simplifies the work of computing the distribution of
the minimum (or maximum.)
"Samik Raychaudhuri" <samik@frKKshKll.org> wrote in message
news:d2aaoc$447$1@news.doit.wisc.edu...
> On 3/28/2005 3:31 PM, kg wrote:
>> I think I understand it a bit better now. Y is the distribution of the
>> minimum values of X1, X2, ..., Xn. I.e. "Y=y" represents that the
>> minimum of X1,...,Xn is y.
>>
>> Is this right?
>>
> P(Y=y)=P(X1>=y, X2>=y, ..., Xn >=y).
> Just curious, are you looking at any particular distributions of X1...Xn?
> Some days back I was looking for the distribution of min(X1...Xn) where
> Xi~N(u_i,sigma_i^2). Couldn't solve it satisfactorily though. Any
> references?
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