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} (X\in A)} is short for {\displaystyle P(\{\omega \in \Omega :X(\omega )\in A\})}, where {\displaystyle \Omega } is the event space and {\displaystyle X(\omega )} is a random variable. {\displaystyle \Pr(A)} notation is used alternatively.

{\displaystyle \mathbb {P} (A\cap B)} or {\displaystyle \mathbb {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 {\displaystyle P(X,Y)}, while joint probability mass function or probability density function as {\displaystyle f(x,y)} and joint cumulative distribution function as {\displaystyle F(x,y)}.

{\displaystyle \mathbb {P} (A\cup B)} or {\displaystyle \mathbb {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. {\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 lowercase letters, e.g. {\displaystyle f(x)}, or {\displaystyle f_{X}(x)}.

Cumulative distribution functions (cdfs) are denoted by uppercase letters, e.g. {\displaystyle F(x)}, or {\displaystyle F_{X}(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)}, or denoted as {\displaystyle S(x)},

In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(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 conditional probability, is the probability of {\displaystyle \textstyle A} given {\displaystyle \textstyle B}, i.e., {\displaystyle \textstyle A} after {\displaystyle \textstyle B} is observed.

Statistics

Greek letters (e.g. θ, β) 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 μ,

the population variance σ²,

the population standard deviation σ,

the population correlation ρ,

the population cumulants κ_r,

{\displaystyle x_{(k)}} is used for the {\displaystyle k^{\text{th}}} order statistic, where {\displaystyle x_{(1)}} is the sample minimum and {\displaystyle x_{(n)}} is the sample maximum from a total sample size n.