Table of Probability Mass Functions

	probability mass function $f(x)$	argument and parameter(s)	Moment-generating function $M_{X}\left( t \right) = E\left( e^{\text{tX}} \right)$	Moments $\mu = E\left( X \right) = \sum_{x}^{}{x \cdot f(x)}$ $\sigma^{2} = \text{Var}\left( X \right) = \sum_{x}^{}{\left( x - \mu \right)^{2} \cdot f\left( x \right)}$	Remark	Corresponding of continuous distributions	R
discrete uniform distribution	$f\left( x \right) = \frac{1}{k}$ for $x = x_{1},\ x_{2},\ \cdots,\ x_{k}$ where $x_{i} \neq x_{j}$ when $i \neq j$	$k = 1,\ 2,\ 3,\ \cdots$	$M_{X}\left( t \right) = \frac{e^{t}\left( 1 - e^{\text{kt}} \right)}{k\left( 1 - e^{t} \right)}$	If $x = 1,\ 2,\ 3,\ \cdots,\ k$ $\mu = \frac{k + 1}{2}$ $\sigma^{2} = \frac{k^{2} - 1}{12}$
Bernoulli distribution	$f\left( x;\ \theta \right) = \theta^{x}\left( 1 - \theta \right)^{1 - x}$ for $x = 0,\ 1$	$x$ (no. of success) $0 \leq \theta \leq 1$	$M_{X}\left( t \right) = 1 + \theta\left( e^{t} - 1 \right)$	$\mu = \theta$ $\sigma^{2} = \theta(1 - \theta)$	binomial distribution with $n = 1$
binomial distribution	$b\left( x;n,\ \theta \right) = \begin{pmatrix}n \ x \ \end{pmatrix}\theta^{x}\left( 1 - \theta \right)^{n - x}$ for $x = 0,\ 1,\ 2,\ \cdots,\ n$	$x$ (no. of success) $n = 1, 2, 3,\cdots$ $0 \leq \theta \leq 1$	$M_{X}\left( t \right) = \left\lbrack 1 + \theta\left( e^{t} - 1 \right) \right\rbrack^{n}$	$\mu = \text{nθ}$ $\sigma^{2} = \text{nθ}(1 - \theta)$	1	Normal distribution of $\mu = \text{nθ}$ $\sigma^{2} = \text{nθ}(1 - \theta)$	[dpqr]binom(x, n, p)
negative binomial distribution	$b^{*}\left( x;k,\ \theta \right) = \begin{pmatrix} x - 1 \ k - 1 \ \end{pmatrix}\theta^{k}\left( 1 - \theta \right)^{x - k}$ for $x = k,\ k + 1,\ k + 2,\ \cdots$			$\mu = \frac{k}{\theta}$ $\sigma^{2} = \frac{k}{\theta}\left( \frac{1}{\theta} - 1 \right)$	2
geometric distribution	$g\left( x;\ \theta \right) = \theta\left( 1 - \theta \right)^{x - 1}$ for $x = 1,\ 2,\ 3,\ \cdots$	$x$ (no. of trials needed to get $k$ success) $k = 1,\ 2,\ 3,\ \cdots$ $0 \leq \theta \leq 1$	$M_{X}\left( t \right) = \frac{\theta e^{t}}{1 - e^{t}\left( 1 - \theta \right)}$	$\mu = \frac{1}{\theta}$ $\sigma^{2} = \frac{1 - \theta}{\theta^{2}}$	negative binomial distribution with $k = 1$	Exponential distribution
hypergeometric distribution	$h\left( x;n,\ N,\ M \right) = \frac{\begin{pmatrix} M \ x \ \end{pmatrix}\begin{pmatrix} N - M \ n - x \ \end{pmatrix}}{\begin{pmatrix} N \ n \ \end{pmatrix}}$ for $x = 0,\ 1,\ 2,\ \cdots,\ n$, $x \leq M$ and $n - x \leq N - M$	$x$ (no. of sampled success elements) $n$ (no. of sampling) $N$ (no. of total elements) $M$ (no. of total success elements)	fairly complex	$\mu = \frac{\text{nM}}{N}$ $\sigma^{2} = \frac{\text{nM}\left( N - M \right)\left( N - n \right)}{N^{2}\left( N - 1 \right)}$
Poisson distribution	$p\left( x;\ \lambda \right) = \frac{\lambda^{x}e^{- \lambda}}{x!}$ for $x = 0,\ 1,\ 2,\ \cdots$	$x$ (no. of successes within a range) $\lambda > 0$	$M_{X}\left( t \right) = e^{\lambda(e^{t} - 1)}$	$\mu = \lambda$ $\sigma^{2} = \lambda$	3		[dpqr]pois(x, lambda)

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1. $f(x;\ \theta) + f(x;\ \theta) + \cdots + f(x;\ \theta)$
   $\rightarrow$ sum of $n$ Bernoulli distributions ↩

2. The number of trials needed to get the $k$th success = $k - 1$ number of successes on $x - 1$ number of trials and $1$ success of Bernoulli distributions ↩

3. Infinite number of Bernoulli distributions within a range
   ($\lambda$ = the rate at which the events occur)
   Approximation of Bernoulli distributions when $n$ is large ↩