Notation in probability and statistics explained

Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.

Probability theory

• Random variables are usually written in upper case Roman letters, such as $X$ or $Y$ and so on. Random variables, in this context, usually refer to something in words, such as "the height of a subject" for a continuous variable, or "the number of cars in the school car park" for a discrete variable, or "the colour of the next bicycle" for a categorical variable. They do not represent a single number or a single category. For instance, if

P(X = x) is written, then it represents the probability that a particular realisation of a random variable (e.g., height, number of cars, or bicycle colour), X, would be equal to a particular value or category (e.g., 1.735 m, 52, or purple), $x$. It is important that $X$ and $x$ are not confused into meaning the same thing. $X$ is an idea, $x$ is a value. Clearly they are related, but they do not have identical meanings.

• Particular realisations of a random variable are written in corresponding lower case letters. For example, $x\_1,x\_2,\; \backslash ldots,x\_n$ could be a sample corresponding to the random variable $X$. A cumulative probability is formally written

P(X\le x) to distinguish the random variable from its realization.^[1]

• The probability is sometimes written

P to distinguish it from other functions and measure P to avoid having to define "P is a probability" and P (X\in A) is short for P(\{\omega \in\Omega: X(\omega) \in A\}) , where \Omega is the event space, X is a random variable that is a function of \omega (i.e., it depends upon \omega ), and \omega is some outcome of interest within the domain specified by \Omega (say, a particular height, or a particular colour of a car). \Pr(A) notation is used alternatively.P (A \cap B) or P [B \cap A] indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as P(X, Y) , while joint probability mass function or probability density function as f(x, y) and joint cumulative distribution function as F(x, y) .P (A \cup B) or P [B \cup A] indicates the probability of either event A or event B occurring ("or" in this case means one or the other or both).

• σ-algebras are usually written with uppercase calligraphic (e.g.

l F for the set of sets on which we define the probability P)

• Probability density functions (pdfs) and probability mass functions are denoted by lowercase letters, e.g.

f(x) , or f _{X(x)} .

• Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g.

F(x) , or F _{X(x)} .

• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:

\overline{F}(x) =1- F(x) , or denoted as S(x) ,

• In particular, the pdf of the standard normal distribution is denoted by $\backslash varphi(z)$, and its cdf by $\backslash Phi(z)$.
• Some common operators:

• $\backslash mathrm[X]$ : expected value of X

• $\backslash operatorname[X]$ : variance of X

• $\backslash operatorname[X,Y]$ : covariance of X and Y
• X is independent of Y is often written

X \perp Y or X \perp\perp Y , and X is independent of Y given W is often written X \perp\perp Y | W orX \perp Y | W style P(A\mid B) , the conditional probability, is the probability of style A given style B

Statistics

• Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).^[2]
• A tilde (~) denotes "has the probability distribution of".
• Placing a hat, or caret (also known as a circumflex), over a true parameter denotes an estimator of it, e.g.,

\widehat{\theta} is an estimator for \theta .

• The arithmetic mean of a series of values $x\_1,x\_2,\; \backslash ldots,x\_n$ is often denoted by placing an "overbar" over the symbol, e.g.

\bar{x} , pronounced "$x$ bar".

• Some commonly used symbols for sample statistics are given below:

\bar{x} ,

• • the sample variance $s^2$,
  • the sample standard deviation $s$,
  • the sample correlation coefficient $r$,
  • the sample cumulants $k\_r$.
• Some commonly used symbols for population parameters are given below:
  • the population mean $\backslash mu$,
  • the population variance $\backslash sigma^2$,
  • the population standard deviation $\backslash sigma$,
  • the population correlation $\backslash rho$,
  • the population cumulants $\backslash kappa\_r$,

x _{(k)} is used for the k ^{th} order statistic, where x _{(1)} is the sample minimum and x _{(n)} is the sample maximum from a total sample size $n$.^[3]

Critical values

The α-level upper critical value of a probability distribution is the value exceeded with probability $\backslash alpha$, that is, the value $x\_\backslash alpha$ such that $F(x\_\backslash alpha)\; =\; 1-\backslash alpha$, where $F$ is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

• $z\_\backslash alpha$ or $z(\backslash alpha)$ for the standard normal distribution
• $t\_$ or $t(\backslash alpha,\backslash nu)$ for the t-distribution with $\backslash nu$ degrees of freedom

{\chi _{\alpha,\nu} }^2 or {\chi} ^{2} (\alpha,\nu) for the chi-squared distribution with $\backslash nu$ degrees of freedomF _{\alpha,\nu _{1,\nu} _{2}} or $F(\backslash alpha,\backslash nu\_1,\backslash nu\_2)$ for the F-distribution with $\backslash nu\_1$ and $\backslash nu\_2$ degrees of freedom

Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. $\backslash bold$.
• Column vectors are usually denoted by boldface lowercase letters, e.g. $\backslash bold$.
• The transpose operator is denoted by either a superscript T (e.g. $\backslash bold^\backslash mathrm$) or a prime symbol (e.g. $\backslash bold\text{'}$).
• A row vector is written as the transpose of a column vector, e.g. $\backslash bold^\backslash mathrm$ or $\backslash bold\text{'}$.

Abbreviations

Common abbreviations include:

• a.e. almost everywhere
• a.s. almost surely
• cdf cumulative distribution function
• cmf cumulative mass function
• df degrees of freedom (also

\nu )

• i.i.d. independent and identically distributed
• pdf probability density function
• pmf probability mass function
• r.v. random variable
• w.p. with probability; wp1 with probability 1
• i.o. infinitely often, i.e.

\{ A _{ni.o.} \} = c ap _{Ncup} _{n\geq} A _{n}

• ult. ultimately, i.e.

\{ A _{n} ult. \} = c up _{Ncap} _{n\geq} A _{n}

See also

• Glossary of probability and statistics
• Combinations and permutations
• History of mathematical notation

External links

• Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.

Notes and References

1. Web site: 2021-08-09 . Calculating Probabilities from Cumulative Distribution Function . 2024-02-26.
2. Web site: 1999-02-13 . Letters of the Greek Alphabet and Some of Their Statistical Uses . 2024-02-26 . les.appstate.edu/.
3. Web site: Order Statistics . 2024-02-26 . colorado.edu.