Notation in probability and statistics

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

Video Notation in probability and statistics

Probability theory

• Random variables are usually written in upper case roman letters: X, Y, etc.
• Particular realizations of a random variable are written in corresponding lower case letters. For example, x₁, x₂, ..., x_n could be a sample corresponding to the random variable X. A cumulative probability is formally written ${\displaystyle P(X\leq x)}$ to differentiate the random variable from its realization.
• The probability is sometimes written ${\displaystyle \mathbb {P} }$ to distinguish it from other functions and measure P so as to avoid having to define " P is a probability" and ${\displaystyle \mathbb {P} (A)}$ is short for ${\displaystyle P(\{\omega :X(\omega )\in A\})}$, where ${\displaystyle \omega }$ is an event and ${\displaystyle X(\omega )}$ a corresponding random variable.
• ${\displaystyle \mathbb {P} (A\cap B)}$ or ${\displaystyle \mathbb {P} [A\cap B]}$ indicates the probability that events A and B both occur.
• ${\displaystyle \mathbb {P} (A\cup B)}$ or ${\displaystyle \mathbb {P} [A\cup B]}$ indicates the probability of either event A or event B occurring ("or" in this case means one or the other or both).
• ?-algebras are usually written with upper case calligraphic (e.g. ${\displaystyle {\mathcal {F}}}$ for the set of sets on which we define the probability P)
• Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
• Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:${\displaystyle {\overline {F}}(x)=1-F(x)}$
• In particular, the pdf of the standard normal distribution is denoted by ?(z), and its cdf by ?(z).
• Some common operators:

• E[X] : expected value of X
• var[X] : variance of X
• cov[X, Y] : covariance of X and Y

• X is independent of Y is often written ${\displaystyle X\perp Y}$ or ${\displaystyle X\perp \!\!\!\perp Y}$, and X is independent of Y given W is often written

${\displaystyle X\perp \!\!\!\perp Y\,|\,W}$ or

${\displaystyle X\perp Y\,|\,W}$

• ${\displaystyle \textstyle P(A\mid B)}$, the posterior probability, is the probability of ${\displaystyle \textstyle A}$ given ${\displaystyle \textstyle B}$, i.e., ${\displaystyle \textstyle A}$ after ${\displaystyle \textstyle B}$ is observed.

Maps Notation in probability and statistics

Statistics

• Greek letters (e.g. ?, ?) are commonly used to denote unknown parameters (population parameters).
• A tilde (~) denotes "has the probability distribution of".
• Placing a hat, or caret, over a true parameter denotes an estimator of it, e.g., ${\displaystyle {\widehat {\theta }}}$ is an estimator for ${\displaystyle \theta }$.
• The arithmetic mean of a series of values x₁, x₂, ..., x_n is often denoted by placing an "overbar" over the symbol, e.g. ${\displaystyle {\bar {x}}}$, pronounced "x bar".
• Some commonly used symbols for sample statistics are given below:
  • the sample mean ${\displaystyle {\bar {x}}}$,
  • the sample variance s²,
  • 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 ?,
  • the population variance ?²,
  • the population standard deviation ?,
  • the population correlation ?,
  • the population cumulants ?_r.

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Critical values

The ?-level upper critical value of a probability distribution is the value exceeded with probability ?, that is, the value x_? such that F(x_?) = 1 - ? where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

• z_? or z(?) for the Standard normal distribution
• t_?,? or t(?,?) for the t-distribution with ? degrees of freedom
• ${\displaystyle {\chi _{\alpha ,\nu }}^{2}}$ or ${\displaystyle {\chi }^{2}(\alpha ,\nu )}$ for the chi-squared distribution with ? degrees of freedom
• ${\displaystyle F_{\alpha ,\nu _{1},\nu _{2}}}$ or F(?,?₁,?₂) for the F-distribution with ?₁ and ?₂ degrees of freedom

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Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g. A.
• Column vectors are usually denoted by boldface lower case letters, e.g. x.
• The transpose operator is denoted by either a superscript T (e.g. A^T) or a prime symbol (e.g. A?).
• A row vector is written as the transpose of a column vector, e.g. x^T or x?.

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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 ${\displaystyle \nu }$)
• 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

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See also

• Glossary of probability and statistics
• Combinations and permutations
• Typographical conventions in mathematical formulae
• History of mathematical notation

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References

• Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", The American Statistician, 19 (3): 12-14, doi:10.2307/2681417, JSTOR 2681417 

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External links

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

Source of the article : Wikipedia