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 to differentiate the random variable from its realization.
• The probability is sometimes written to distinguish it from other functions and measure P so as to avoid having to define “P is a probability” and is short for, where is the event space and is a random variable. notation is used alternatively.
• or indicates the probability that events A and B both occur. The joint probability distribution of random variables X and Y is denoted as, while joint probability mass function or probability density function as and joint cumulative distribution function as.
• or indicates the probability of either event A or event B occurring.
• σ-algebras are usually written with uppercase calligraphic
• Probability density functions and probability mass functions are denoted by lowercase letters, e.g., or.
• Cumulative distribution functions are denoted by uppercase letters, e.g., or.
• Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:, or denoted as,
• In particular, the pdf of the standard normal distribution is denoted by φ, and its cdf by Φ.
• Some common operators:
• X is independent of Y is often written or, and X is independent of Y given W is often written
• , the conditional probability, is the probability of given, i.e., after is observed.

  Statistics

• Greek letters are commonly used to denote unknown 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., is an estimator for.
• The arithmetic mean of a series of values x₁, x₂,..., x_n is often denoted by placing an "overbar" over the symbol, e.g., pronounced "x bar".
• Some commonly used symbols for sample statistics are given below:
• *the sample mean,
• *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 μ,
• *the population variance σ²,
• * the population standard deviation σ,
• *the population correlation ρ,
• *the population cumulants κ_r,
• is used for the order statistic, where is the sample minimum and is the sample maximum from a total sample size n.

  Critical values

• z_α or z for the standard normal distribution
• t_α,ν or t for the t-distribution with ν degrees of freedom
• or for the chi-squared distribution with ν degrees of freedom
• or F for the F-distribution with ν₁ and ν₂ degrees of freedom

  Linear algebra

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

  Abbreviations

• a.e. almost everywhere
• a.s. almost surely
• cdf cumulative distribution function
• cmf cumulative mass function
• df degrees of freedom
• 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.
• ult. ultimately, i.e.