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)

Replying to a 16-year-old post doesn't make a lot of sense, but I've just landed on that page for the first time because of Quantling's comment below. In case someone is wondering: there is indeed a well-developed theory of power series with non-negative coefficients. A central result in that context is the Vivanti–Pringsheim theorem. For more on this, see e.g. Flajolet's "Analytic combinatorics". Malparti (talk) 19:36, 13 March 2026 (UTC)

Why limit as z approaches 1 from below?

If we plug in z = 1 without taking any limits, the power series evaluates to the sum of the probability masses. That is simply 1 without needing any limits. As far as I can tell the talk of limits can be removed from the article. (If I have this wrong; please provide an example where G(z=1) does not converge absolutely. Thank you —Quantling (talk | contribs) 17:51, 13 March 2026 (UTC)

You're correct. Note however that it is necessary to take the limit for derivatives; and that the notation that is currently (namely $1^{-}$) has to be introduced somewhere (so fixing the article will likely take a bit more work than just removing that limit). Malparti (talk) 19:32, 13 March 2026 (UTC)

Thank you for your quick response. I'm not so sure that the limit helps with derivatives. Sure, with a probability distribution such as ⁠$p(x)\propto {\frac {1}{x^{\alpha }}}$⁠ for ⁠$x\geq 1$⁠ and ⁠$\alpha >1$⁠, the value ⁠$\operatorname {E} [x^{\beta }]$⁠ will diverge unless ⁠$\beta <\alpha -1$⁠. When ⁠$\beta <\alpha -1$⁠, we get ⁠$G^{(\beta )}(1^{-})=G^{(\beta )}(1)$⁠, both finite. When ⁠$\beta \geq \alpha -1$⁠, I am thinking that both ⁠$G^{(\beta )}(1^{-})$⁠ and ⁠$G^{(\beta )}(1)$⁠ will diverge "to ⁠$+\infty$⁠". That is, for this family of probability distributions, we can replace all instances of ⁠$G^{(\beta )}(1^{-})$⁠ with ⁠$G^{(\beta )}(1)$⁠. Is there a probability distribution that you know of where the probability generating function differs for ⁠$G^{(\beta )}(1^{-})$⁠ and ⁠$G^{(\beta )}(1)$⁠? —Quantling (talk | contribs) 20:22, 13 March 2026 (UTC)

There is no such example, in the following sence:

• since $X$ is nonnegative, $\mathbb {E} (X)$ always exists in $\mathbb {R} \cup \{+\infty \}$;
• since $G$ is differentiable on $(0,1)$ and $G'$ is nondecreasing (because the coefficients of the power series are nonnegative), $\textstyle G'(1^{-}):=\lim _{s\uparrow 1}G'(s)$ always exists in $\mathbb {R} \cup \{+\infty \}$;
• by monotone convergence, $G'(1^{-})=\mathbb {E} (X)$ in $\mathbb {R} \cup \{+\infty \}$.

The issue is simpler than that: it's just that $G'$ isn't well-defined in 1, in the usual sense (i.e. as a real or complex derivative) : for this, you'd need $G$ to be defined on a neighborhood of 1.

Now, I agree that if you interpret $G'$ as a real derivative, then $\textstyle \lim _{s\uparrow 1}G'(s)$ exists and will give you the same result as what you get by "plugging in" $z=1$ in the formal derivative of the power series... So you could very well make up your own convention and say that $G'(1):=\textstyle \lim _{s\uparrow 1}G'(s)$. However:

1. That is a non-standard convention, so non-ideal for Wikipedia;
2. I'd advice against it, because it tends to hide the fact that something special could happen in 1; for instance, if the radius of convergence of the power series is 1, then 1 is guaranteed to be a singularity. Also, this "natural" convention is likely more arbitrary than it seems, because I assume — I haven't thought this through so I might very well be wrong here — that it should be possible to find probability generating functions with a singularity in 1 where you can get different limits of $G'(z_{n})$ as $z_{n}\to 1$ from within the disc of convergence. If that is indeed the case, then the convention $G'(1)=\textstyle \lim _{s\uparrow 1}G'(s)$ wouldn't make sense.

Malparti (talk) 23:39, 13 March 2026 (UTC)

Ah, I see, I think. I am interpreting ⁠$G^{(n)}(z)$⁠ as a formal power series created by differentiating the formal power series ⁠$G(z)$⁠, term by term, a total of ⁠$n$⁠ times. With that it is okay to simply evaluate the resulting formal power series with ⁠$z=1$⁠. However if we interpret ⁠$G(z)$⁠ as a (non-formal) function then differentiating at the point ⁠$z=1$⁠ even once requires that ⁠$G(z)$⁠ be well defined in a neighborhood of that point.

When we define ⁠$G(z)$⁠ we say that it is a generating function. I think if we're clear that by "differentiation" we mean "as a formal power series, term by term" then we can replace occurrences of ⁠$G^{(n)}(1^{-})$⁠ with ⁠$G^{(n)}(1)$⁠ and simplify the discussion accordingly. However, without such a clarifying sentence, readers might easily assume that we mean differentiation in the normal sense for functions; keeping the ⁠$G^{(n)}(1^{-})$⁠ and related discussion would cover that interpretation.

Perhaps it boils down to which is more accessible to the reader: (a) formal differentiation of a formal power series term by term vs. (b) computing the desired values as limits. —Quantling (talk | contribs) 18:15, 14 March 2026 (UTC)

The probability generating function is typically introduced as $s\mapsto \mathbb {E} (s^{X})$ for $s\in (-R,R)$ or $z\mapsto \mathbb {E} (z^{X})$ for $|z|<R$ — depending how familiar the audience is complex analysis.

Now, $\mathbb {E} (z^{X})$ is not a formal power series: it is defined entirely differently, and then proved to be numerically equal to the series $\textstyle \sum _{n}p_{n}z^{n}$, inside its disk of convergence. Of course, given this equality, it makes perfect sense if you want to define the probability generating function as a formal power series, rather than as a function — and in fact, this standard in analytic combinatorics. But it's not in probability theory (at least at the undergrad level).

As a result, in probability courses the standard approach is to say that the expected value (in $\mathbb {R} \cup \{+\infty \}$) of $X$ is obtained as "the limit of $G'(s)$ as $s\uparrow 1$ on the real line" (if $G$ is viewed as a real function, then it's also quite common to phrase things in terms of its left-derivative); not as "what you get by evaluating the corresponding power series at $z=1$". So I think this article should to stick to that.

Cheers,

Malparti (talk) 23:28, 14 March 2026 (UTC)

I think of ⁠$\mathbb {E} (z^{X})$⁠ as a formal power series equal to ⁠$\sum _{n=0}^{\infty }p_{n}z^{n}$⁠ ... I guess my background is analytic combinatorics. But because you indicate that this expectation is instead an ordinary function in undergraduate probability texts, that's good enough for me. Thanks! —Quantling (talk | contribs) 19:35, 15 March 2026 (UTC)

@Malparti Maybe I gave in too easily. After all, a probability generating function is a generating function, right? And according to that Wikipedia article, a generating function is a formal series, not an ordinary function, right? I conclude that we are formally differentiating a formal series, not ordinary differentiation of a function that is defined by an ordinary series. So sure, we should be sure to explain what we mean by formal differentiation of a formal power series (in a brief sentence, with a wikilink to an appropriate article), but once we've done it, I am still thinking it is more appropriate to use ⁠$G^{(n)}(1)$⁠ than ⁠$G^{(n)}(1^{-})$⁠. —Quantling (talk | contribs) 02:59, 16 March 2026 (UTC)

@Quantling "a probability generating function is a generating function, right? And according to that Wikipedia article, a generating function is a formal series, not an ordinary function, right?" → A <blabla> is whatever you define it to be in whatever field you're working in. In combinatorics, it is natural to view generating functions as formal power series, because they can have a zero radius of convergence. That's not the case for PGFs: they carry a lot of extra structure so it makes sense to view them as more constrained objects than formal power series.

My point is not that defining $G'(1)$ as "the derivative of the formal power series, evaluated in 1" wouldn't make sense — it does make sense, in the sense that it doesn't create any obvious contradiction (although, as I said, I personally think it is a bad idea, because any notation that hides potential pitfalls under the rug is a recipe for errors, in practice). My point is: that's not the way things are done in probability theory, at least not in the textbooks I've personally read. So, if you want to keep arguing on this, you need to take the time to look at sources.

Here is a (short) list of some of the most influential textbooks in probability theory, combining some old classics with more modern texts: Feller, Billingsley, Shiryaev, Chung, Williams, Durrett. I don't have all of them on my computer, and I do have a few other ones:

• Feller (1968) views $G$ as a real function, defined where the series converges; after discussing the convention $\mathbb {E} (X)=+\infty$, sets $\textstyle G'(1):=\lim _{s\uparrow }G'(s)$; see Chapter XI.
• Kallenberg (1997) only mentions the PGF in passing, and defines it as a real function on $[0,1]$, and doesn't really say anything about; see page 61.
• Durrett (2019) also doesn't spend a lot of time on the PGF, and also defines in as a real function on $[0,1]$ and is careful to say "We may have φ(s) = ∞ when s > 1 so we have to work carefully" and then goes on to use $\textstyle \lim _{s\uparrow 1}\phi '(s)$.
• I don't think Chung and Billingsley discuss PGFs.

I don't think the burden should be on me to look at other sources.

As a Wikipedia contributor, your contributions should not be to give your opinion on which convention makes more sense when working with PGFs, but rather to see what convention sources use, and summarize them here. Malparti (talk) 10:06, 16 March 2026 (UTC)

Thank you for As a Wikipedia contributor, your contributions should not be to give your opinion on which convention makes more sense when working with PGFs, but rather to see what convention sources use, and summarize them here. I agree and I revert back to yielding to using the limit formulations. —Quantling (talk | contribs) 14:35, 16 March 2026 (UTC)