Probability/Local Manual of Style

Purpose of this book
The difficulty level of this book should be similar to that of university-level probability course. In particular, and related advanced topic should  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.

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.
 * letters (possibly with subscripts): or  ;
 * letters (possibly with subscripts): or  in sets;
 * $$A\cup B$$: the of $$A$$ and $$B$$;
 * $$A\cap B$$: the of $$A$$ and $$B$$;
 * $$A\setminus B$$: the of $$B$$ in $$A$$
 * $$A\subseteq B$$: $$A$$ is a of $$B$$;
 * $$A\subsetneq B$$: $$A$$ is a of $$B$$;
 * $$S^c$$: the (absolute) of $$S$$;
 * $$U$$: a set;
 * $$|S|$$: the of $$S$$;
 * $$\mathcal P(S)$$: the of $$S$$;
 * $$\binom{n}{r}$$: the indexed by $$n$$ and $$r$$;
 * $$\Omega$$: a ;
 * $$\mathcal F$$: an ;
 * $$\mathbb P$$: the (function);
 * $$A\perp\!\!\!\perp B$$: $$A$$ and $$B$$ are ;
 * $$F$$: a ;
 * $$f$$: a or ;
 * $$\operatorname{supp}(X)$$: the of $$X$$;
 * $$\operatorname{Binom}(n,p)$$: the distribution with $$n$$ independent Bernoulli trials with success probability $$p$$;
 * $$\operatorname{Ber}(p)$$: the distribution with one Bernoulli trial with success probability $$p$$;
 * $$\operatorname{Pois}(\lambda)$$: the distribution with rate parameter $$\lambda$$;
 * $$\operatorname{Geo}(p)$$: the distribution with success probability $$p$$;
 * $$\operatorname{NB}(k,p)$$: the negative binomial distribution (number of failures before $$k$$th successes) with success probability $$p$$;
 * $$\operatorname{HypGeo}(N,K,n)$$: the distribution with population size $$N$$ containing $$K$$ objects of type 1, $$N-K$$ objects of another type, and $$n$$ objects drawn;
 * $$\operatorname{FD}(\mathbf x,\mathbf p)$$: the distribution with vector $$\mathbf x$$ and probability vector $$\mathbf p$$;
 * $$\operatorname{D}\mathcal U\{x_1,\dotsc,x_n\}$$: the distribution;
 * $$\mathcal{U}[a,b]$$: the distribution over the interval $$[a,b]$$;
 * $$\operatorname{Exp}(\lambda)$$ : the distribution with rate parameter $$\lambda$$;
 * $$\operatorname{Gamma}(\alpha,\lambda)$$: the distribution with shape parameter $$\alpha$$ and rate parameter $$\lambda$$;
 * $$\operatorname{Beta}(\alpha,\beta)$$: the distribution with shape parameters $$\alpha$$ and $$\beta$$;
 * $$\operatorname{Cauchy}(\theta)$$: the distribution with location parameter $$\theta$$ (with scale parameter 1);
 * $$\mathcal{N}(\mu,\sigma^2)$$: the distribution with mean $$\mu$$ and variance $$\sigma^2$$;
 * $$\chi^2_\nu$$: the distribution with $$\nu$$ degrees of freedom;
 * $$t_\nu$$: the distribution with $$\nu$$ degrees of freedom;
 * $$F_{\nu_1,\nu_2}$$: the $$F$$-distribution with $$\nu_1$$ and $$\nu_2$$ degrees of freedom;
 * $$\operatorname{Multinom}(n,\mathbf p)$$: the  distribution with $$n$$ trials and probability vector $$\mathbf p$$.
 * $$\mathcal{N}_k(\boldsymbol\mu,\boldsymbol\Sigma)$$: the $$k$$-dimensional distribution with mean vector $$\boldsymbol\mu$$ and covariance matrix $$\boldsymbol\Sigma$$;
 * $$\mathbb E[X]$$ (or $$\mu_X$$): the of $$X$$;
 * $$\operatorname{Var}(X)$$ (or $$\sigma^2_X$$): the of $$X$$;
 * $$\sigma_X$$: the of $$X$$;
 * $$\operatorname{Cov}(X,Y)$$: the of $$X$$ and $$Y$$;
 * $$\rho(X,Y)$$ (or $$\rho_{XY}$$) : the of $$X$$ and $$Y$$;
 * Bold letters (e.g. $$\mathbf X$$, and possibly with subscript): ;
 * Bold letters (e.g. $$\mathbf x$$, and possibly with subscript): ;
 * $$\mathbf x^T$$: the of $$\mathbf x$$;
 * $$\mathbf x\cdot\mathbf y$$: the of $$\mathbf x$$ and $$\mathbf y$$.

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 $$\nu$$ (stands for 'nu', possibly with subscript).


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

Conventions

 * Use casing for  (called chapter) titles, and use  casing for section titles.
 * Use LaTeX (instead of HTML) for math-related variables, formulas, notations etc., to ensure consistency in appearance.
 * Use $$$$ for math ;
 * Use $$$$ for math (i.e. formulas on its own line);


 * Use quizzes (if possible) for exercises.
 * Try to use mnemonic notations (if possible). E.g., $$S$$ for a set, $$t$$ for time, etc.

Templates

 * Use nav as navigation template. Put two nav's for each subpage, one at top, one at bottom, and enclose each of them by, 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.