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Concept
Notation in probability and statistics
Summary
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.
Random variables are usually written in upper case roman letters: , , etc.
Particular realizations 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 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 ("or" in this case means one or the other or both).
σ-algebras are usually written with uppercase 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 lowercase letters, e.g. , or .
Cumulative distribution functions (cdfs) 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:
expected value of X
variance of X
covariance of X and Y
X is independent of Y is often written or , and X is independent of Y given W is often written
or
the conditional probability, is the probability of given , i.e., after is observed.
Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
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