List of generating functions

Couldn’t find any list of generating functions for distributions on the Internet so I started compiling my own. (They will be filled in as they appear in the wild)

Definition

We use the simpler notation for the generating function of $X\in F$

$\begin{equation} g_{X}(t) \overset{\text{def}}{=} g_{F}(t) \end{equation}$

And by definition the moment generating function is

$\begin{equation} \label{eq:generatingfunctiondefinition} g_{X}(t) \overset{def}{=} E[t^X] = \sum t^k p_X(k) \end{equation}$

The random variables $X$ are in all cases $\mathbb{Z}_{\ge 0}$

Sum

$X_i$ are independent random variables.

$\begin{equation} g_{(\sum X_i)}(t) \overset{\eqref{eq:generatingfunctiondefinition}}{=} E[t^{\sum X_i}] = E\left[\prod t^{X_i} \right] = \prod E[t^{X_i}] \overset{\eqref{eq:generatingfunctiondefinition}}{=} \prod g_{X_i}(t) \end{equation}$

Scale

$\begin{equation} g_{aX}(t) \overset{\eqref{eq:generatingfunctiondefinition}}{=} E[t^{aX}] = E[(t^a)^X] \overset{\eqref{eq:generatingfunctiondefinition}}{=} g_{X}(t^a) \end{equation}$

Translation

$\begin{equation} g_{X+b}(t) \overset{\eqref{eq:generatingfunctiondefinition}}{=} E[t^{X+b}] = E[t^Xt^b] = E[t^X]t^b \overset{\eqref{eq:generatingfunctiondefinition}}{=} g_{X}(t)t^b \end{equation}$

Evaluate $p_X(k)$ (inverse)

$\begin{align} \frac{d^k}{k!dt^k}g_X(t=0) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \frac{d^k}{k!dt^k}E[t^X]\Bigg|_{t=0} \nonumber \\ &= \frac{d^k}{k!dt^k}\sum_{k'} t^{k'} p_X(k') \Bigg|_{t=0} \nonumber \\ &= \frac{1}{k!}\sum_{k'} p_X(k') \frac{d^k}{dt^k} t^{k'} \Bigg|_{t=0} \nonumber \\ &= \frac{1}{k!}\sum_{k'} p_X(k') \frac{k'!}{(k'-k)!}t^{k'-k} \Bigg|_{t=0} \nonumber \\ &= \frac{1}{k!}\sum_{k'} p_X(k') \frac{k'!}{(k'-k)!}0^{k'-k} \nonumber \\ &= \frac{1}{k!}\sum_{k'} p_X(k') \frac{k'!}{(k'-k)!}\delta(k'-k) \nonumber \\ &= \frac{1}{k!}\sum_{k'=k} p_X(k') \frac{k'!}{(k'-k)!}\delta(k'-k) \nonumber \\ &= \frac{1}{k!} p_X(k) \frac{k!}{0!} \cdot 1 \nonumber \\ &= p_X(k) \label{eq:evaluate_generating} \end{align}$

Inverse of a polynomial $g_X(t) = \sum\limits_n c_n t^n$

$\begin{align} p_X(k) &\overset{\eqref{eq:evaluate_generating}}{=} \frac{d^k}{k!dt^k} g_X(t)\Bigg|_{t=0,\, g_X(t) = \sum\limits_n c_n t^n} \nonumber \\ &= \frac{d^k}{k!dt^k} \sum\limits_n c_n t^n \Bigg|_{t=0} \nonumber \\ &= \sum\limits_n \frac{d^k}{k!dt^k} c_n t^n \Bigg|_{t=0} \nonumber \\ &= \sum\limits_n \frac{1}{k!} \frac{n!}{(n-k)!} c_n t^{n-k} \Bigg|_{t=0} \nonumber \\ &= \sum\limits_n \frac{1}{k!} \frac{n!}{(n-k)!} c_n 0^{n-k} \nonumber \\ &= \sum\limits_{n=k} \frac{1}{k!} \frac{n!}{(n-k)!} c_n 0^{n-k} \nonumber \\ &= \frac{1}{k!} \frac{k!}{(k-k)!} c_k 0^{k-k} \nonumber \\ &= \frac{1}{k!} \frac{k!}{0!} c_k 0^0 \nonumber \\ &= c_k \end{align}$

Identities

$\begin{align} g_{X}(1) = E[1^X] = 1 \end{align}$

Dirac

$\begin{align} g_{\delta(a)}(t) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \sum\limits_{k\in\mathbb{Z}_{\ge 0}} \delta(a)(k) t^k \nonumber \\ &= \sum\limits_{k\in\mathbb{Z}_{\ge 0}} \delta_a(k) t^k \nonumber \\ &= t^a \end{align}$

Poission

$\begin{align} g_{\text{Po}(m)}(t) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \sum\limits_{k\in\mathbb{Z}_{\ge 0}} \text{Po}(m)(k)t^k \nonumber \\ &= \sum\limits_{k\in\mathbb{Z}_{\ge 0}} e^{-m}\frac{m^k}{k!}t^k \nonumber \\ &= e^{-m}\sum\limits_{k\in\mathbb{Z}_{\ge 0}} \frac{(mt)^k}{k!} \nonumber \\ &= e^{-m}e^{mt} \nonumber \\ &= (e^{t-1})^m = e^{m(t-1)} \end{align}$

Binomial

$\begin{align} g_{\text{Bin}(n, p)}(t) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \sum\limits_{k=0}^n \text{Bin}(n, p)(k)t^k \nonumber \\ &= \sum\limits_{k=0}^n \binom{n}{k} p^k(1-p)^{n-k}t^k \nonumber \\ &= \sum\limits_{k=0}^n \binom{n}{k} (pt)^k(1-p)^{n-k} \nonumber \\ &= (pt + (1-p))^n \label{eq:binomial} \end{align}$

Bernoulli

$\begin{align} g_{\text{Be}(p)} &= g_{\text{Bin}(n=1, p)} \nonumber \\ &\stackrel{\eqref{eq:binomial}}{=} pt + (1-p) \end{align}$

Geometric

$\begin{align} g_{\text{Ge}(p)}(t) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \sum\limits_{k\in\mathbb{Z}_{\ge 0}} \text{Ge}(p)(k) t^k \nonumber \\ &= \sum\limits_{k\in\mathbb{Z}_{\ge 0}} p(1-p)^k t^k \nonumber \\ &= p\sum\limits_{k\in\mathbb{Z}_{\ge 0}} ((1-p)t)^k \nonumber \\ &= \frac{p}{1-(1-p)t}, \quad |(1-p)t| < 1 \end{align}$

First Success

$\begin{align} g_{\text{Fs}(p)}(t) &\overset{\eqref{eq:generatingfunctiondefinition}}{=} \sum\limits_{k\in\mathbb{Z}_{\ge 1}} \text{Fs}(p)(k) t^k \nonumber \\ &= \sum\limits_{k\in\mathbb{Z}_{\ge 1}} p(1-p)^{k-1} t^k \nonumber \\ &= \frac{p}{1-p}\sum\limits_{k\in\mathbb{Z}_{\ge 1}} ((1-p)t)^k \nonumber \\ &= \frac{p}{1-p}\left(\sum\limits_{k\in\mathbb{Z}_{\ge 0}} ((1-p)t)^k - 1\right) \nonumber \\ &= \frac{p}{1-p}\left(\frac{1}{1-(1-p)t} - 1\right),\quad |(1-p)t|<1 \nonumber \\ &= \frac{p}{1-p} \frac{1-(1-(1-p)t)}{1-(1-p)t} \nonumber \\ &= \frac{pt}{1-(1-p)t},\quad |(1-p)t|<1 \end{align}$