Talk:Negative hypergeometric distribution

The PDF formula refers to the parameter n, which is undefined. This is probably a typo. --Erel Segal (talk) 15:15, 23 March 2015 (CET)

Typo, indeed. It should be $N$, not $n$. Thank you. Would you like to fix it yourself? --Boris Tsirelson (talk) 18:08, 23 March 2015 (CET)

The equation was an image I could not edit, so I replaced it with a latex equation - I hope I did this right. --Erel Segal (talk) 20:31, 24 March 2015 (CET)

Yes, this is one of our problems; you could look at Help:HowTo EoM, or just see some new articles made in TeX. Anyway, I did it a bit nicer. Boris Tsirelson (talk) 08:17, 25 March 2015 (CET)

Yes, this looks much nicer. While we are at it: I did not understand why this formula is correct. Can you please add an intuitive explanation? Also, I didn't understand the formula at the bottom, connecting the negative-hypergeometric with the hypergeometric. Can you explain this too? --Erel Segal (talk) 08:25, 25 March 2015 (CET)

Wow... not now, maybe later. Boris Tsirelson (talk) 08:30, 25 March 2015 (CET)

The PMF formula (*) is suspicious. If m<N, then k+m-N<k, so the top-left factor is zero! --Erel Segal (talk) 19:23, 25 March 2015 (CET)

Some sources available:

• planetmath;
• N. Balakrishnan, V.B. Nevzorov, "A primer on statistical distributions", Wiley 2004 (see page 103);
• Shaughnessy;
• Mansuri

Boris Tsirelson (talk) 20:45, 26 March 2015 (CET)

Hmmm... After rewriting the three formulas in the last three sources in terms of EoM I got, respectively, \[ \frac{ {k+m-1 \choose k}{N-m-k \choose M-m} } { {N \choose M} } \, ; \] \[ \frac{ {M \choose k-1}{N-M \choose m} } { {N \choose m+k-1} } \cdot \frac{ M-k+1 }{ N-m-k+1 } \; , \] \[ \frac{ {k+m-1 \choose k-1}{N-m-k \choose M-m} } { {N \choose M} } \, . \] I am puzzled. Boris Tsirelson (talk) 21:43, 26 March 2015 (CET)

In PlanetMath, the meaning of the parameters is only partially explained. But probably their $B$ and $W$ are our $M$ and $N-M$, and their $b$ and $x$ are our $m$ and $k$. If so, then their formula is identical to the formula from Balakrishnan and Nevzorov. At least, a hope to see two identical formulas!.. Boris Tsirelson (talk) 21:55, 26 March 2015 (CET)

Good news! Two more sources give the same as Balakrishnan and Nevzorov:

• G.K. Miller, S.L. Fridell, "A forgotten discrete distribution? Reviving the negative hypergeometric model", The American Statistician 2007 61:4, 347-350.
• E.F. Schuster, W.R. Sype, "On the negative hypergeometric distribution", Int. J. Math. Educ. Sci. Technol. 1987 18:3, 453-459.

Boris Tsirelson (talk) 22:36, 26 March 2015 (CET)

So, now I feel sure enough for correcting the article (even though I did not check the calculations myself). Thanks to Erel Segal for the good question. Boris Tsirelson (talk) 22:42, 26 March 2015 (CET)

Thanks for sorting this out! --Erel Segal (talk) 05:46, 27 March 2015 (CET)