Wikipedia:Reference desk/Archives/Mathematics/2023 May 31

= May 31 =

Are data min and max quartiles? How?
In trying to learn about quartiles, I read the article Quartile. I thought I understood it, until I got to this part "Along with the minimum and maximum of the data (which are also quartiles) [...]". What does this mean? I don't understand how the min and max of the data could also be a quartile. Can someone explain it to me? Maybe its just a typo and I don't know enough about this subject to know that. This is not a homework assignment, but it is related to school: my school has started reporting what appears to be the Five-number summary of the assignment scores. RudolfRed (talk) 00:35, 31 May 2023 (UTC)
 * Quartile in that article means a boundary point of the four quarters. There are three internal boundary points: between the first quarter and the second quarters; between the second quarter and the third; and between the third quarter and the fourth.  But there are also the boundary points on the outside: the min of the data set is the lower boundary of the first quarter; the max of the data set is the upper boundary of the fourth quarter.  So technically those are quartiles as well, though that doesn't really come up.--2404:2000:2000:8:FE:DFAD:D729:3ACD (talk) 02:40, 31 May 2023 (UTC)
 * Thanks for the explanation. RudolfRed (talk) 03:28, 31 May 2023 (UTC)

Finding the CDF of the sum of n discrete uniform random variables
I'm having trouble with Generatingfunctionology chapter 1 exercise 10g. Pastebin: https://mathb.in/75444. I got the required PGF but I don't know how to expand it to get the required CDF. I can do it the way I detail in the pastebin or I can write $$G_{S_n}(t)=\frac {t^n} {m^n} \left( \sum_{i=0}^{m-1} t^i \right)^n \left( \sum^\infty_{k=0} t^k \right)$$ but in both cases I need to find the Cauchy product between a finite expansion and an infinite series, and for the latter method I first need to raise the finite expansion to the nth power, and I don't know how to actually do the computation in both cases.

There is a paper by Caiado and Rathie (2007) [1] about raising a finite expansion to the nth power and the resulting polynomial coefficients but they use a notation $${\binom {n} {j}}_{k+1}$$ (supposedly from Comtet 1974) that I don't follow so I don't understand their findings.

[1] Caiado, C.C. and Rathie, P.N., 2007, May. Polynomial coefficients and distribution of the sum of discrete uniform variables. In Eighth Annual Conference of the Society of Special Functions and their Applications, Pala, India, Society for Special Functions and their Applications. Manc2124 (talk) 20:41, 31 May 2023 (UTC)


 * Comtet 1974 can be downloaded from the Internet Archive here. These polynomial coefficients are defined, with a different notation, in Exercise 18 on page 77. If I made no mistake, Calado and Rathie's $${\binom nj}_{k+1}$$ is the same as $$\begin{pmatrix}n,k{+}1\\j\end{pmatrix}$$ in Comtet's notation. --Lambiam 08:53, 1 June 2023 (UTC)
 * Thanks, worked it out: https://mathb.in/75452 82.33.131.214 (talk) 14:06, 1 June 2023 (UTC)