Notation in Probability and Statistics - Probability Theory

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.
• or indicates the probability that events A and B both occur.
• or 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 upper case calligraphic (e.g. 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).
• In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
• Some common operators:

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