A Distribution of Range of Samples taken from N(0,1)

  • A
  • Thread starter Thread starter uart
  • Start date Start date
uart
Science Advisor
Messages
2,797
Reaction score
21
TL;DR Summary
Is there a name for the probability distribution of the range of a sample (size n) taken from a standard normal population?
I recently came across a distribution called the "Studentized Range" (for example, implemented as ptukey(x,n,dof) and qtukey(x,n,dof) in the R software package). Essentially it's the distribution of the range (max sample value - min sample value), for a sample (size n) taken from a student_t distribution (degrees of freedom dof).

Was just curious if there is a named distribution for the same thing except with the sample being taken from N(0,1) instead of student_t. Basically I was wondering if it's a distribution that's already implemented in various stats packages of if it just remains unnamed and unloved?

I know it can be approximated by just using a large dof in existing "Studentized Range" implementations like ptukey(), but wondering if it already exists as a named and tabulated distribution in it's own right.
 
Last edited:
Physics news on Phys.org
Just updating this thread. As far as I call tell this distribution is not commonly named or tabulated. It can however be computed from the following equation, which is a general expression for the distribution of the range (W) of a sample of random variables with distribution,density F(x),f(x).

$$F_W(W) = n \int_{-\infty}^\infty f(u) [F(u+W) - F(u)]^{n-1} du$$

I have now implemented this (crudely) in gnu Octave and it gives very good agreement with the asymptotic behavior (for large dof) of the ptukey/qtukey functions mentioned above. Can post the .m files if anyone is interested, but be warned the implementation is extremely crude and inefficient, as it was just to test that the method worked.

Reference: https://demonstrations.wolfram.com/DistributionOfTheSampleRangeOfContinuousRandomVariables/
 
Last edited:
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.
Back
Top