# Probability/Local Manual of Style

## Purpose of this book

The difficulty level of this book should be similar to that of first university-level probability course. In particular, measure theory and related advanced topic should not be included in this book. Instead, they should be included in the Probability Theory wikibook (for measure-theoretic probability), or Measure Theory wikibook (for measure theory itself).

Applications of probability can be included briefly, but are not the main focus of this wikibook.

## Some notations and abbreviations

### Notations

In some occasions, these notations may have different meanings compared with those stated in the following. The words explaining the meaning of the following notations in the actual content take precedence.

• ${\displaystyle {\overset {\text{def}}{=}}}$ : equals by definition
• CAPITAL letters (possibly with subscripts): sets [1] or random variables [2];
• small letters (possibly with subscripts): variables or elements in sets;
• ${\displaystyle A\cup B}$ : the union of ${\displaystyle A}$  and ${\displaystyle B}$ ;
• ${\displaystyle A\cap B}$ : the intersection of ${\displaystyle A}$  and ${\displaystyle B}$ ;
• ${\displaystyle A\setminus B}$ : the relative complement of ${\displaystyle B}$  in ${\displaystyle A}$
• ${\displaystyle A\subseteq B}$ : ${\displaystyle A}$  is a subset of ${\displaystyle B}$ ;
• ${\displaystyle A\subsetneq B}$ : ${\displaystyle A}$  is a proper subset of ${\displaystyle B}$ ;
• ${\displaystyle S^{c}}$ : the (absolute) complement of ${\displaystyle S}$ ;
• ${\displaystyle U}$ : a universal set;
• ${\displaystyle \#(S)}$ : the cardinality of ${\displaystyle S}$ ;
• ${\displaystyle {\mathcal {P}}(S)}$ : the power set of ${\displaystyle S}$ ;
• ${\displaystyle {\binom {n}{r}}}$ : the binomial coefficient indexed by ${\displaystyle n}$  and ${\displaystyle r}$ ;
• ${\displaystyle \Omega }$ : a sample space;
• ${\displaystyle {\mathcal {F}}}$ : a event space;
• ${\displaystyle \mathbb {P} }$ : the probability (function);
• ${\displaystyle A\perp \!\!\!\perp B}$ : ${\displaystyle A}$  and ${\displaystyle B}$  are independent;
• ${\displaystyle F}$ : a cumulative distribution function;
• ${\displaystyle f}$ : a probability mass or probability density function;
• ${\displaystyle \operatorname {supp} (X)}$ : the support of ${\displaystyle X}$ ;
• ${\displaystyle \operatorname {Binom} (n,p)}$ : the binomial distribution with ${\displaystyle n}$  independent Bernoulli trials with success probability ${\displaystyle p}$ ;
• ${\displaystyle \operatorname {Ber} (p)}$ : the Bernoulli distribution with one Bernoulli trial with success probability ${\displaystyle p}$ ;
• ${\displaystyle \operatorname {Pois} (\lambda )}$ : the Poisson distribution with rate parameter ${\displaystyle \lambda }$ ;
• ${\displaystyle \operatorname {Geo} (p)}$ : the geometric distribution with success probability ${\displaystyle p}$ ;
• ${\displaystyle \operatorname {NB} (k,p)}$ : the negative binomial distribution (number of failures before ${\displaystyle k}$ th successes) with success probability ${\displaystyle p}$ ;
• ${\displaystyle \operatorname {HypGeo} (N,K,n)}$ : the hypergeometric distribution with population size ${\displaystyle N}$  containing ${\displaystyle K}$  objects of type 1, ${\displaystyle N-K}$  objects of another type, and ${\displaystyle n}$  objects drawn;
• ${\displaystyle \operatorname {FD} (\mathbf {x} ,\mathbf {p} )}$ : the finite discrete distribution with vector ${\displaystyle \mathbf {x} }$  and probability vector ${\displaystyle \mathbf {p} }$ ;
• ${\displaystyle \operatorname {D} {\mathcal {U}}\{x_{1},\dotsc ,x_{n}\}}$ : the discrete uniform distribution;
• ${\displaystyle {\mathcal {U}}[a,b]}$ : the uniform distribution over the interval ${\displaystyle [a,b]}$ ;
• ${\displaystyle \operatorname {Exp} (\lambda )}$  [3]: the exponential distribution with rate parameter ${\displaystyle \lambda }$ ;
• ${\displaystyle \operatorname {Gamma} (\alpha ,\lambda )}$ : the gamma distribution with shape parameter ${\displaystyle \alpha }$  and rate parameter ${\displaystyle \lambda }$ ;
• ${\displaystyle \operatorname {Beta} (\alpha ,\beta )}$ : the beta distribution with shape parameters ${\displaystyle \alpha }$  and ${\displaystyle \beta }$ ;
• ${\displaystyle \operatorname {Cauchy} (\theta )}$ : the Cauchy distribution with location parameter ${\displaystyle \theta }$  (with scale parameter 1);
• ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$ : the normal distribution with mean ${\displaystyle \mu }$  and variance ${\displaystyle \sigma ^{2}}$ ;
• ${\displaystyle \chi _{\nu }^{2}}$ : the chi-squared distribution with ${\displaystyle \nu }$  degrees of freedom;
• ${\displaystyle t_{\nu }}$ : the Student's ${\displaystyle t}$  distribution with ${\displaystyle \nu }$  degrees of freedom;
• ${\displaystyle F_{\nu _{1},\nu _{2}}}$ : the ${\displaystyle F}$ -distribution with ${\displaystyle \nu _{1}}$  and ${\displaystyle \nu _{2}}$  degrees of freedom;
• ${\displaystyle \operatorname {Multinom} (n,\mathbf {p} )}$ : the multinomial distribution with ${\displaystyle n}$  trials and probability vector ${\displaystyle \mathbf {p} }$ .
• ${\displaystyle {\mathcal {N}}_{k}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$ : the ${\displaystyle k}$ -dimensional multivariate normal distribution with mean vector ${\displaystyle {\boldsymbol {\mu }}}$  and covariance matrix ${\displaystyle {\boldsymbol {\Sigma }}}$ ;
• ${\displaystyle \mathbb {E} [X]}$  (or ${\displaystyle \mu _{X}}$ ): the mean of ${\displaystyle X}$ ;
• ${\displaystyle \operatorname {Var} (X)}$  (or ${\displaystyle \sigma _{X}^{2}}$ ): the variance of ${\displaystyle X}$ ;
• ${\displaystyle \sigma _{X}}$ : the standard deviation of ${\displaystyle X}$ ;
• ${\displaystyle \operatorname {Cov} (X,Y)}$ : the covariance of ${\displaystyle X}$  and ${\displaystyle Y}$ ;
• ${\displaystyle \rho (X,Y)}$  (or ${\displaystyle \rho _{XY}}$ ) : the correlation coefficient of ${\displaystyle X}$  and ${\displaystyle Y}$ ;
• Bold CAPITAL letters (e.g. ${\displaystyle \mathbf {X} }$ , and possibly with subscript): random vectors;
• Bold small letters (e.g. ${\displaystyle \mathbf {x} }$ , and possibly with subscript): vectors;
• ${\displaystyle \mathbf {x} ^{T}}$ : the transpose of ${\displaystyle \mathbf {x} }$ ;
• ${\displaystyle \mathbf {x} \cdot \mathbf {y} }$ : the dot product of ${\displaystyle \mathbf {x} }$  and ${\displaystyle \mathbf {y} }$ .

### Abbreviations

• no.: number;
• r.v.: random variable;
• cdf: cumulative distribution function;
• pmf: probability mass function;
• pdf: probability density function;
• s.d.: standard deviation;
• df: degrees of freedom;
• It is usually denoted by ${\displaystyle \nu }$  (stands for 'nu', possibly with subscript).
• i.i.d: independent and identically distributed;
• mgf: moment generating function;
• CLT: Central Limit Theorem.

## Conventions

• Use title casing for subpage (called chapter) titles, and use sentence casing for section titles.
• Use LaTeX (instead of HTML) for all math-related variables, formulas, notations etc., to ensure consistency in appearance[4].
• Use  for inline math[5];
• Use <math display=block>[/itex] for display math (i.e. formulas on its own line);
• Use quizzes (if possible) for exercises.
• Try to use mnemonic notations (if possible). E.g., ${\displaystyle S}$  for a set, ${\displaystyle t}$  for time, etc. [6]

## Templates

1. usually the first few letters in alphabetical order , e.g. ${\displaystyle A,B}$  and ${\displaystyle C}$ , or ${\displaystyle S}$  (mnemonic for set)
2. usually the last few letters in alphabetical order), e.g. ${\displaystyle X,Y}$  and ${\displaystyle Z}$
3. It is not ${\displaystyle \exp(\lambda )}$ .
4. For numbers, they can be just typed out.
5. This makes the symbol bigger than that using <math display=inline>[/itex], and thus is clearer.
6. However, conventional notations should take precedenece. E.g., we should use ${\displaystyle \mu }$ , instead of ${\displaystyle m}$ , for mean.