Probability Notation

Statisticians love technical language. When keywords and specific names become too verbose, they invent a notation.

See also some σ algebra notation, Bayesian notation, joint probability notation, conditional probability notation, expected value notation, and conditional expectation notation.

Distribution

A random variable is distributed by some function. This relationship is notated with a tidle, as in X ~ Bernoulli.

Expected Values

The expected value of X is notated as E[X].

Probability mass functions

A discrete random variable is distributed by a probability mass function (PMF). This is typically notated as p(X=x). Sometimes the function is named P (capitalized) or Pr instead.

For Bernoulli-distributed random variables, because the only possible values are 0 and 1, and because the 0 term evaluated out of most equations, a shorthand notation is commonly used. p(X=1) = p(X).

Sometimes a probability function is notated with a subscript to emphasize what random variable it described. For example, p_X(x) = p(X=x).

Probability density functions

A continuous random variable is distributed by a probability density function (PDF).

While such a function may be expressed as p(X=x) (or p_X(x)), it isn't possible to evaluate this function at a single value. See CDFs instead.

Cumulative distribution functions

The probability that a random variable takes a value equal or less than x is given by a cumulative distribution function (CDF). For discrete variables, this is a summation of the PMF for all values from 0 to x. For continuous variables, this is the integral of the PDF from 0 to x.

If a CDF is named F then it is evaluated like F_X(x) = P(X <= x).

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