# Probability/Local Manual of Style

Appearance

It is encouraged to follow this manual when editing to ensure consistency. Nevertheless, be bold when editing. |

## Purpose of this book

[edit | edit source]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

[edit | edit source]### Notations

[edit | edit source]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.

*CAPITAL*letters (possibly with subscripts):*sets*^{[1]}or*random variables*^{[2]};*small*letters (possibly with subscripts):*variables*or*elements*in sets;- : the
*union*of and ; - : the
*intersection*of and ; - : the
*relative complement*of in - : is a
*subset*of ; - : is a
*proper subset*of ; - : the (absolute)
*complement*of ; - : a
*universal*set; - : the
*cardinality*of ; - : the
*power set*of ; - : the
*binomial coefficient*indexed by and ; - : a
*sample space*; - : an
*event space*; - : the
*probability*(function); - : and are
*independent*; - : a
*cumulative distribution function*; - : a
*probability mass*or*probability density function*; - : the
*support*of ; - : the
*binomial*distribution with independent Bernoulli trials with success probability ; - : the
*Bernoulli*distribution with one Bernoulli trial with success probability ; - : the
*Poisson*distribution with rate parameter ; - : the
*geometric*distribution with success probability ; - : the negative binomial distribution (number of failures before th successes) with success probability ;
- : the
*hypergeometric*distribution with population size containing objects of type 1, objects of another type, and objects drawn; - : the
*finite discrete*distribution with vector and probability vector ; - : the
*discrete uniform*distribution; - : the
*uniform*distribution over the interval ; -
^{[3]}: the*exponential*distribution with rate parameter ; - : the
*gamma*distribution with shape parameter and rate parameter ; - : the
*beta*distribution with shape parameters and ; - : the
*Cauchy*distribution with location parameter (with scale parameter 1); - : the
*normal*distribution with mean and variance ; - : the
*chi-squared*distribution with degrees of freedom; - : the
*Student's*distribution with degrees of freedom; - : the -distribution with and degrees of freedom;
- : the
*multinomial*distribution with trials and probability vector . - : the -dimensional
*multivariate normal*distribution with mean vector and covariance matrix ; - (or ): the
*mean*of ; - (or ): the
*variance*of ; - : the
*standard deviation*of ; - : the
*covariance*of and ; - (or ) : the
*correlation coefficient*of and ; **Bold***CAPITAL*letters (e.g. , and possibly with subscript):*random vectors*;**Bold***small*letters (e.g. , and possibly with subscript):*vectors*;- : the
*transpose*of ; - : the
*dot product*of and .

### Abbreviations

[edit | edit source]- 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 (stands for 'nu', possibly with subscript).

- i.i.d: independent and identically distributed;
- mgf: moment generating function;
- CLT: Central Limit Theorem.

## Conventions

[edit | edit source]- 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 <math></math> for
*inline*math^{[5]}; - Use <math display=block></math> for
*display*math (i.e. formulas on its own line);

- Use <math></math> for

- Use quizzes (if possible) for exercises.
- Try to use mnemonic notations (if possible). E.g., for a set, for time, etc.
^{[6]}

## Templates

[edit | edit source]- Use {{nav}} as navigation template. Put two {{nav}}'s for each subpage, one at top, one at bottom, and enclose each of them by <noinclude></noinclude>, so that they do not appear at the print version.
- Use {{colored em}} for emphasis.
- Use {{colored definition}}, {{colored proposition}}, {{colored theorem}}, {{colored remark}}, {{colored proof}}, {{colored example}}, {{colored exercise}}, {{colored corollary}} and {{colored lemma}} for writing definitions, propositions, etc. ('type declarations' for paragraphs).
- Put {{BookCat}} at the bottom of each subpage.

- ↑ usually the
*first*few letters in alphabetical order , e.g. and , or (mnemonic for set) - ↑ usually the
*last*few letters in alphabetical order), e.g. and - ↑ It is
*not*. - ↑ For numbers, they can be just typed out.
- ↑ This makes the symbol bigger than that using <math display=inline></math>, and thus is clearer.
- ↑ However, conventional notations should take precedenece. E.g., we should use , instead of , for mean.