Talk:Skellam distribution

Use of mu and lambda

Is there a good reason why the Poisson distribution is given here with mu as a parameter, and on its own page in terms of lambda? It would probably help if this were to be tidied up, I think. Johnprovis 09:55, 12 October 2007 (UTC)

Mu is more widely used as a symbol denoting the mean of a process. Lambda is often used in fields such as chemical kinetics to denote the average rate of a chemical process and in the common example of nuclear decay used to illustrate the Poisson distribution this is the symbol used. It is probably best to leave the articles as they are since each symbol is most familiar in their separate contexts. dweomer —Preceding unsigned comment added by 76.23.19.40 (talk) 18:03, 27 April 2009 (UTC)

Order?

In line: "where I_k(z) is the modified Bessel function of the first kind." I think the k defines the order of the used modified Bessel function and therefore the correct phrase would be "where I_k(z) is the modified Bessel function of the k-th kind." SveFinBioInf 12:17, 27 April 2010 (CEST)

No, it is correct as it stands. Check out Bessel function. PAR (talk) 12:22, 27 April 2010 (UTC)

Ah OK you are right. I was confused by the k, which stands for the order of the modified Bessel function of the first kind. Is this correct? Thanks! SveFinBioInf 15:56, 17 June 2010 (CEST)

Yes, thats right. More complete would be "I_k(z) is the modified Bessel function of the first kind of order k". PAR (talk) 23:33, 17 June 2010 (UTC)

The line about |k|

As much as I know the distribution is defined by

$p(k;\mu _{1},\mu _{2})=\Pr\{K=k\}=e^{-(\mu _{1}+\mu _{2})}\left({\mu _{1} \over \mu _{2}}\right)^{k/2}I_{|k|}(2{\sqrt {\mu _{1}\mu _{2}}})$

However, it does say in the first paragraph Since k is an integer we have that I_k(z)=I_|k|(z). I tried to check if it's true but it didn't seem so obvious to me, or maybe the line isn't clear? Maybe there should be a link to a source with a proof to this note? — Preceding unsigned comment added by 82.81.84.146 (talk) 13:36, 12 July 2017 (UTC)

Median

Can anyone explained why the median is simply given as "N/A"? I was under the impression that all probability distributions had at least one median. I cannot find anything on the median of the distribution, but that's not to say it doesn't exist. --Ipatrol (talk) 05:01, 2 April 2020 (UTC)

I haven't found any literature on this either. But after doing some work with the Python "scipy.stats.skellam" module, I believe the median is $u_{1}-u_{2}$, round half towards zero (i.e. $-2.5$ is $-2$). I'm not well-versed in probability though. Hope someone can (dis)prove it or point to some reference. – TrunghaiTĐN (talk) 14:48, 18 April 2020 (UTC)

That's approximately it, but as with the Poisson median, it's not quite accurate. Take mu1 = 0.7, mu2 = 0.1, difference is 0.6, which would round up towards 1, but the actual median is still 0. Aaron Denney (talk) 04:12, 5 November 2023 (UTC)