Notation in probability and statistics

Probability theory

• Random variables are usually written in upper case Roman letters, such as or 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 is written, then it represents the probability that a particular realisation of a random variable, X, would be equal to a particular value or category,. It is important that and are not confused into meaning the same thing. is an idea, 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, could be a sample corresponding to the random variable . A cumulative probability is formally written to distinguish the random variable from its realization.
• The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and is short for, where is the event space, is a random variable that is a function of , and is some outcome of interest within the domain specified by . 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, and its cdf by.
• 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''

  Statistics

• Greek letters are commonly used to denote unknown parameters.
• Some commonly used symbols for population or distribution parameters are given below:
• *the population mean,
• *the population variance,
• * the population standard deviation ',
• *the population correlation ',
• *the population cumulants ',
• 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 is often denoted by placing an "overbar" over the symbol, e.g., pronounced " bar".
• Some commonly used symbols for sample statistics are given below:
• *the sample mean,
• *the sample variance,
• * the sample standard deviation ',
• *the sample correlation coefficient ,
• *the sample cumulants.
• is used for the order statistic, where is the sample minimum and is the sample maximum from a total sample size.

  Critical values

• or for the standard normal distribution
• or for the t-distribution with degrees of freedom
• or for the chi-squared distribution with degrees of freedom
• or for the F-distribution with and degrees of freedom

  Linear algebra

• Matrices are usually denoted by boldface capital letters, e.g..
• Column vectors are usually denoted by boldface lowercase letters, e.g. '.
• 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. ' or .

  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.