Talk:Probability-generating function

The listed p.g.f for negative binomial is wrong. There should be no $z$ on the numerator, i.e.

$\left({\frac {p}{1-qz}}\right)^{r}$

It all depends on whether you are after the number of trials required to achieve n successes (which is at least n) or the number of failures before the nth success (which may be 0). From a quick check of the article geometric distribution, what you suggest above is correct for the latter definition, while the formula in the article is consistent with the text, which refers to the former. Ben Cairns 03:11, 2 Mar 2005 (UTC).

Actually, it is correct that there should be no $z$ in the numerator. Having it in the numerator is appropriate in the case where the distribution is on $1,2,...$, but the text here has it on $0,1,2,...$ for which there is clearly a nonzero probability of $0$ Since the probability of $0$ is found by plugging in $z=0$ it's easy to see that we cannot have $z$ in the numerator. Joelmiller (talk) 03:40, 23 August 2017 (UTC)

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What on earth does the notation G(1-) mean? It's used all over this article, and I've never seen it anywhere before. Is it supposed to be G(-1)? Or G(1)? Or is it just a typo? Could someone clarify/correct it please.

I hadn't seen it before but it is defined in the article. It is the limit as z->1 of G(z). Since this is used so much it is useful to have a shorthand though I have not seen this one before. --Richard Clegg 18:15, 4 June 2006 (UTC)

I'm not sure, but doesn't it mean the limit of G(x) as x approaches 1 from below? PAR 18:25, 4 June 2006 (UTC)

I think it is just a "typo". BM

The relation to "moment generating function"

Some Wikipedia articles about discrete random variables such as "Geometric distribution" use the term "moment generating function" instead of "probability generating function". So it would be useful to state that these are equivalent ideas in such cases.

edit; I don't mean "equivalent ideas". I should say "related ideas". My thought is that it would be useful to tell a reader if he can get the formula for the probability generating function by looking at the formula for the moment generating function since the articles favor giving the moment generating function.

Can't this be merged into moment generating function? I can't see why this needs a separate article. —3mta3 (talk) 13:48, 3 May 2009 (UTC)

Examples

We need to take a close look at the didactical merits of the examples, deleting some if necessary. -Zahlentheorie (talk) 13:32, 8 July 2010 (UTC)

Clarifications needed

• What is "to make available the well-developed theory of power series with non-negative coefficients" in the lead referring to? ... the article seems to have nothing relevant (and a link to another article would be good if there is something relevant). The content at Series (mathematics)##Non-negative terms does not indicate anything particularly useful. And there seems nothing in Abel's theorem that makes specific use of "positive coefficients".
• The paragraph including "the probability-generating function of the difference of two independent random variables" must be extending the types of random variables concerned to include ones which can take negative values, but these are excluded from the present definition.
• The treatment of the G(1^-) stuff needs to be rethought, particularly for the derivatives. Can a general reader be expected to interpret the content here as meaning that the mean does not exist if the the limit for the derivative does not exist?

Melcombe (talk) 12:08, 20 September 2010 (UTC)