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Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

has some commonly used conventions of its own, in addition to standard mathematical notation

Mathematical notation

Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...

and mathematical symbols

Table of mathematical symbols

This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

Probability theory

Random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s 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
Random sample
In statistics, a sample is a subject chosen from a population for investigation; a random sample is one chosen by a method involving an unpredictable component...

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
Sigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

are usually written with upper case calligraphic
Calligraphy
Calligraphy is a type of visual art. It is often called the art of fancy lettering . A contemporary definition of calligraphic practice is "the art of giving form to signs in an expressive, harmonious and skillful manner"...

(e.g. for the set of sets on which we define the probability P)
Probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

s (pdfs) and probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

s are denoted by lower case letters, e.g. f(x).
Cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

s (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[X] : expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of X
var[X] : variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

of X
cov[X, Y] : covariance
Covariance
In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

of X and Y

Statistics

Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
An estimate
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....

of a parameter is often denoted by placing a caret
Caret
Caret usually refers to the spacing symbol ^ in ASCII and other character sets. In Unicode, however, the corresponding character is , whereas the Unicode character named caret is actually a similar but lowered symbol: ....

over the corresponding symbol, e.g. , pronounced "theta hat".
Some commonly used symbols for sample statistics are given below:
- the sample mean ,
- the sample variance s²,
- 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 correlation
  Pearson product-moment correlation coefficient
  In statistics, the Pearson product-moment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
  
  ρ,
- the population cumulant
  Cumulant
  In probability theory and statistics, the cumulants κn of a probability distribution are a set of quantities that provide an alternative to the moments of the distribution. The moments determine the cumulants in the sense that any two probability distributions whose moments are identical will have...
  
  s κ_r.
The arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

of a set of numbers x₁, x₂, ..., x_n is denoted by , pronounced "x bar".

Critical values

The α-level upper critical value

Critical value

-Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...

of a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

is the value exceeded with probability α, that is, the value x_α such that F(x_α) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:

z_α or z(α) for the Standard normal distribution
t_α,ν or t(α,ν) for the t-distribution with ν degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
or for the chi-squared distribution with ν degrees of freedom
or F(α,ν₁,ν₂) for the F-distribution with ν₁ and ν₂ degrees of freedom

Linear algebra

Matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

are usually denoted by boldface capital letters, e.g. A.
Column vectors are usually denoted by boldface lower case letters, e.g. x.
The transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...

operator is denoted by either a superscript T (e.g. A^T) or a prime symbol
Prime (symbol)
The prime symbol , double prime symbol , and triple prime symbol , etc., are used to designate several different units, and for various other purposes in mathematics, the sciences and linguistics...

(e.g. A′).
A row vector is written as the transpose of a column vector, e.g. x^T or x′.

Abbreviations

Common abbreviations include:

a.e. almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
a.s. almost surely
Almost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
cdf cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
cmf cumulative mass function
df degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

(also )
i.i.d. independent and identically distributed
Independent and identically distributed random variables
In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent....
pdf probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...
pmf probability mass function
Probability mass function
In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...
r.v. random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
w.p. with probability; wp1 with probability 1

External links

Earliest Uses of Symbols in Probability and Statistics, maintained by Jeff Miller.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Probability theory

Statistics

Critical values

Linear algebra

Abbreviations

See also

External links