Wikipedia:Reference desk/Archives/Mathematics/2022 March 10

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 10

Mixed Poisson Distribution

In the German article for Mixed Poisson Distribution (see e.g. here) states:

The Mixed Poisson Distribution is a generalization of the Poisson Distribution.

I cannot see why this should be true. The PMF of a Mixed Poisson Distribution with PDF g(x) is $f(k)={\frac {1}{k!}}\int _{0}^{\infty }x^{k}e^{-x}g(x)\mathrm {d} x$ which is not Poisson distibuted in general. Greetings Bigbossfarin (talk) 14:35, 10 March 2022 (UTC)

The English article Mixed Poisson distribution states the same, so I do not get why you are referring to the German Wikipedia. We have an article named Compound probability distribution, which is another term for mixed distribution – although some authors make a somewhat trivial distinction. In general, the compound distribution obtained from composing a distribution of type X and a distribution of type Y is itself of neither of the two types. The mixed Poisson distribution is called a generalization of the usual Poisson distribution because the latter can be retrieved from the former as a special case, namely by choosing $\pi (\lambda )=\delta (\lambda -\lambda _{0}),$ in which $\delta$ denotes the Dirac delta function, representing a random variable with a one-point distribution.  --Lambiam 16:51, 10 March 2022 (UTC)

Dear @Lambiam: I was writing the English article a few hours ago, and I was already guessing, that there could be a way with Dirac delta. I will think about it. Greetings Bigbossfarin (talk) 18:37, 10 March 2022 (UTC)

Using the variable names of Compound probability distribution § Definition, the "outer" parametrized distribution $F$ can always be retrieved by using a shifted Dirac delta distribution for the "inner" parameter distribution $G$. So the fact that this mixed Poisson distribution is "a generalization" of the Poisson distribution is not an interesting observation. I see we also have an article with the title Compound Poisson distribution, in which it is the "inner" parameter distribution that is Poisson.  --Lambiam 07:15, 11 March 2022 (UTC)

Mathematics using templates

How do I insert the divided value of {{NUMBEROFEDITS}} / {{NUMBEROFACTIVEUSERS}} into a page? Hcoder3104☭ (💬) 18:10, 10 March 2022 (UTC)

By using the parser functions extension #expr; see Help:Extension:ParserFunctions § #expr at Mediawiki. For example, {{#expr:10/7 round 5}} produces 1.42857. You'll need to filter out the commas first using {{formatnum:...|R}}; otherwise #expr chokes on the commas. For example, while {{NUMBEROFEDITS}} may produce 1,071,649,631, by using {{formatnum:{{NUMBEROFEDITS}}|R}} you get the string 1071649631.  --Lambiam 19:26, 10 March 2022 (UTC)