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

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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 $f(k)=\frac{1}{k!}\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^k{e}^{-x}g(x)\mathrm{d}x$ 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 Template:Ping 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 Template:Section link, 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 Template:Mono; see Help:Extension:ParserFunctions § #expr at Mediawiki. For example, Template:Nowrap produces 1.42857. You'll need to filter out the commas first using Template:Mono; otherwise Template:Mono chokes on the commas. For example, while Template:Mono may produce Template:Mono, by using Template:Mono you get the string Template:Mono.  --Lambiam 19:26, 10 March 2022 (UTC)