View Full Version : The Minimum of n Independent Normal Distributions

07-09-2010, 02:15 PM
This is a very good article, on the subject of the distribution of the minimum of several random variables, by Joshua Hill:


Although this problem is very simple to solve using random numbers, the finding of a formula is very difficult and it's the first time I saw a treatment of the subject, although I 've been looking for it.

Can you help me understand more about this ?
In particular, how does the computational method using "Mathematika" described at the end of Joshua Hill's work ?
How can I make an algorithm for it, outside the "Mathematika" environment ?
Or is there some other simpler approximation to the formula ?

07-09-2010, 02:40 PM
Just a quick note... it's "Mathematica" and you don't need the quotes every time.

Note that the title you gave the thread is more correct than your first line in that the paper talks about the specific case of normal random variables. I only bring it up because you mentioned in the other thread about the max of random variables that they're related and they are indeed related but that was talking in general for random variables with a common CDF that differ only by a location parameter.

Simulations are indeed easier to do than the mathematics behind it. The abstract manipulations of order statistics isn't too horrible if you're dealing with the min or the max but once you actually have a specific distribution to work with things can be ugly if you actually want to integrate.

07-10-2010, 03:47 AM
It's a cdf rather than pdf, and it solves the problem of maximum also (for unocorrelated variables), as maximum = -minimum.
I don't know how to paste the integral here but it looks to me that the
part involving x, the test parameter, factors out of the integration.

So it reduces to something like:

p(x) = A . EXP ( - B . (X - M ) ^ 2 )

where A, B are problem specific constants (but not global constants) and M is the theoretical mean
of the variable x.
Can this be true ?

That's a fairly recent paper (published in May 10, 2010).