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Appendix:Glossary of probability and statistics

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The following is a glossary of terms related to probability and statistics.

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atomic event 
Another name for elementary event.


A sample not being representative of the population.
The difference between the expected value of an estimator and the true value.
Bayesian probability 
(or "personal probability") .. ?
Bayesian inference 


categorical variable 
A nominal variable.
conditional distribution 
Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X (written "Y | X") is the probability distribution of Y when X is known to be a particular value.
conditional probability 
the probability of some event A, assuming event B. Conditional probability is written P(A|B), and is read "the probability of A, given B".
, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables (one can use it to quantify, for example, how shoe size and height are correlated in the population). An example is the Pearson product-moment correlation coefficient, which is found by dividing the covariance of the two variables by the product of their standard deviations. Independent variables have a correlation of 0.
between two random variables X and Y, with expected values <math>E(X)=\mu</math> and <math>E(Y)=\nu</math> is defined as the expected value of random variable <math>(X - \mu) (Y - \nu)</math>, and is written <math>\operatorname{cov}(X, Y)</math>. It is used for measuring correlation.
A subjective estimate of probability.


data set 
A sample and the associated data points.
data point 
A typed measurement - it can be a boolean value, a real number, a vector (in which case it's also called a data vector), etc.
distribution function 
The function that gives the probability distribution of a random variable. It cannot be negative, and its integral on the probability space is equal to 1.


eclectic probability 
elementary event 
(or atomic event) is an event with only one element. For example, when pulling a card out of a deck, "getting the jack of spades" is an elementary event, while "getting a king or an ace" is not.
A function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. The mean can be used as an estimator.
expected value 
(or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff ("value"). Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. For example, the expected value of a six-sided die roll is 3.5. The concept is similar to the mean. The expected value of random variable X is typically written E(X) or <math>\mu</math> (mu).
A subset of the sample space, to which a probability can be assigned. For example, on rolling a die, "getting a five or a six" is an event (with a probability of one third if the die is fair).


frequency probability 


generating function 


independence or statistical independence 
Two events are independent if the outcome of one does not affect that of the other (for example, getting a 1 on one die roll does not affect the probability of getting a 1 on a second roll). Similarly, when we assert that two random variables are independent, we intuitively mean that knowing something about the value of one of them does not yield any information about the value of the other.
interval variable 
A variable with the features of ordinal variable, such that equal differences between its values represent equivalent intervals.


joint distribution 
Given two random variables X and Y, the joint distribution of X and Y is the probability distribution of X and Y together.
joint probability 
the probability of two events occurring together. The joint probability of A and B is written <math>P(A \cap B)</math> or <math>P(A, \ B).</math>


A measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent modestly-sized deviations.


likelihood function 
(or just likelihood) is a conditional probability function considered a function of its second argument with its first argument held fixed. For example, imagine pulling a numbered ball with the number k from a bag of n balls, numbered 1 to n. Then you could describe a likelihood function for the random variable N as the probability of getting k given that there are n balls : the likelihood will be 1/n for n greater or equal to k, and 0 for n smaller than k. Unlike a probability distribution function, this likelihood function will not sum up to 1 on the sample space.


marginal distribution 
given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y.
marginal probability 
The probability of an event, ignoring any information about other events. The marginal probability of A is written P(A). Contrast with conditional probability.
Of a random variable is its expected value. The mean (or sample mean of a data set is just the average value.
moment about the mean 
mutual independence 
A collection of events is mutually independent if for any subset of the collection, the joint probability of all events occurring is equal to the product of the joint probabilities of the individual events. Think of the result of a series of coin-flips. This is a stronger condition than pairwise independence.


nominal variable 
A variable with values whose order is insignificant.


ordinal variable 
A variable with values whose order is significant.


pairwise independence 
a pairwise independent collection of random variables is a set of random variables any two of which are independent.
Can be a population parameter, a distribution parameter, an unobserved parameter (all the same ?). Often written θ.
prior probability 
population or statistical population 
is a set of entities about which statistical inferences are to be drawn, often based on random sampling. One can also talk about a population of measurements or values.
population parameter 
See statistical parameter.
posterior probability 
probability density 
Is used to describe probability in a continuous probability distribution. For example, you can't say that the probability of a man being six feet tall is 20%, but you can say he has 20% of chances of being between five and six feet tall. Probability density is given by a probability density function. Contrast with probability mass.
probability density function 
gives the probability distribution for a continuous random variable.
probability distribution 
A function that gives the probability of all elements in a given space. (-> see that page for a list of different distributions)
probability measure 
gives the probability of events in a probability space.
probability space 
A sample space over which a probability measure has been defined.
random function 
probability theory 


random variable 
A quantity whose values are random and to which a probability distribution is assigned, such as the possible outcomes of a roll of a dice.
discrete random variable 
continuous random variable 
random vector (or multivariate random variable
a vector whose components are random variables on the same probability space.
ratio variable 
A variable with the features of interval variable, whose any two values have meaningful ratio, making the operations of multiplication and division meaningful.


sample or statistical sample 
That part of a population which is actually observed.
sample space 
The set of possible outcomes of an experiment. For example, the sample space for rolling a six-sided die will be {1, 2, 3, 4, 5, 6}.
A process of selecting observations to obtain knowledge about a population. There are many methods to choose on which sample to do the observations.
sampling distribution 
The probability distribution, under repeated sampling of the population, of a given statistic.
A measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew (right-skewed) if the higher tail is longer and negative skew (left-skewed) if the lower tail is longer (confusing the two is a common error).
standard deviation 
The most commonly used measure of statistical dispersion. It is the square root of the variance, and is generally written <math>\sigma</math> (sigma).
standardized moment 
The result of applying a statistical algorithm to a data set. It can also be described as an observable random variable.
statistical inference 
Inference about a population from a random sample drawn from it or, more generally, about a random process from its observed behavior during a finite period of time.
statistical dispersion 
(also called statistical variability) is a measure of how diverse some data is. It can be expressed by the variance or the standard deviation.
statistical parameter 
is a parameter that indexes a family of probability distributions.
of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. The variance of random variable X is typically designated as <math>\operatorname{var}(X)</math>, <math>\sigma_X^2</math>, or simply <math>\sigma^2</math>.

See also