This standard specifies commonly used mathematical statistics terms. This standard applies to mathematical statistics terms involved in various standards and technical documents. It should also be used as a reference for mathematical statistics terms involved in various research and technical reports and works. GB/T 3358.1-1993 Statistical terminology Part 1: General statistical terms GB/T3358.1-1993 standard download decompression password: www.bzxz.net
This standard specifies commonly used mathematical statistics terms. This standard applies to mathematical statistics terms involved in various standards and technical documents. It should also be used as a reference for mathematical statistics terms involved in various research and technical reports and works.

National Standard of the People's Republic of China
Statistical Terms
Part I : Terms for general statistics1 Subject content and scope of application
This standard specifies the commonly used mathematical statistics terms. GB/T3358.1-·93
Substitute GR3358:82
This standard applies to the mathematical statistics terms involved in various standards and technical documents, and should also be used as a reference for the mathematical statistics terms involved in various research and technical reports and works.
2 Probability Terms
2.1 Probability
A real number that measures the probability of a random event occurring, and its value is between 0 and 1. Note: The probability of a random event can be regarded as the stable value of the frequency of the event when the experiment is repeated under the same conditions, or as the degree of belief in the occurrence of the event:
2.2 [One-dimensional] random variable
[univariate] random variable, variate A variable whose value depends on the test results and has a certain probability distribution. 2.3 [-dimensional] probability distribution [univariate] probability distribution A function that gives the probability of a random variable taking any given value or taking a value in any given set, and the probability of a random variable taking a value in its entire variation range is 1.
2.4 [-dimensional distribution function (univariate) distribution function The probability that a random variable X is less than or equal to a real number, it is a function of: F(α) = P(X ≤T)
Note: F(r) is right continuous. Sometimes Fr)P(Xn) is also defined as the distribution function. In this case, F(r) is left continuous. 2.5 Continuous random variables and probability density function continuous random variable and [probability'density function
If the distribution function F(x) of a random variable can be expressed as the integral of a non-negative function (): F(α) =
f(t)dt
then the random variable is called a continuous random variable, and F(α) is called its [probability] density function. Any density function f() must satisfy
f(r)dx = 1
Note: The probability distribution of a continuous random variable is called a continuous distribution. 2.6 Discrete random variables and probability functionsdiscreterandom variables and probability functions probabilityfunction A random variable X that can only take a finite or countable number of values ​​(·2,) is called a discrete random variable. The function that gives the probability of X taking possible values ​​is called a probability function:
Any probability function must satisfy =1.
GB/T3358.193
p, P(X -r,).i - 1,2,...
Note: The probability distribution of a discrete random variable is called a discrete distribution. 2.7-dimensional random variable k-dimensional random variable The value depends on the test results and has a certain dimensional probability distribution table dimensional vector: X- (X,,X2,--,X.).
2.8-dimensional probability distribution k-dimensional probability distribution A function that gives the probability of a -dimensional random variable taking values ​​in any given set in k-dimensional space. The probability distribution of the D-dimensional (three-2) random variables is also called the joint distribution of the D-dimensional random variables.
Note: (The real-valued function
F(r;r,,r) P(X, sai+X r.,X a) on the D-dimensional space is called the joint distribution of the D-dimensional random variable X-(X,X..X.). ②If the distribution function F() of the D-dimensional random variable can be expressed as the integral of a non-negative function r.): Fa+ - . taddt
then the D-dimensional random variable is called the D-dimensional continuous random variable. The distribution of the D-dimensional continuous random variable ri,s, is called the D-dimensional continuous distribution.
is called its probability gate density function. The D-dimensional continuous random variable can only take finite or listable values ​​(x2,,) and the D-dimensional random variable is called the D-dimensional discrete random variable. The function that gives the probability of it taking each possible value (i\a) is called the D-dimensional probability function. p, = P(X, - x,\,X = x.),i - I.2,..The distribution of the D-dimensional discrete random variable is called the D-dimensional discrete distribution. 2.9 Marginal distribution The joint distribution of P components of a π-dimensional random variable is called the π-dimensional marginal distribution of the distribution of the π-dimensional random variable. Example: A π-dimensional random variable (X, Y, Z) contains: -three 2D marginal distributions, namely the distributions of (X, Y), (X, Z) and (Y, Z); -three 2D marginal distributions, namely the distributions of X, Y and Z. 2.10 Conditional distribution The joint probability distribution of P components of a π-dimensional random variable under the condition that another P components take given values. 2.11 Independence
If two groups of random variables have a joint probability distribution, and the conditional distribution of any one group does not change with the value of the other group, then they are said to be independent. Otherwise, they are said to be dependent. Note: (i) If two groups of random variables (X, X) and (Yi, …, Y,) are independent, then for any (,) and (y.), F(rirmyi+.yh) = Fi(a+,arn)F(yi.**+y), where F(.rm) and F(,) are the distribution functions of (X,.X) and (Y...Y.), respectively, and F(..r..) is their joint distribution number.
Similar formulas also hold for density functions and probability functions. The above concept can be extended to the case of group random variables. 2.12 Quantile quantile
For a random variable, a real number that satisfies the conditions (X<)≤ and PX≤)= is called the quantile of its distribution.
Note: (1) Quantiles may not be unique -
2) and m. are called upper and lower quartiles respectively. 2.73 Median
The 0.5 quantile of a random variable or its probability distribution. 2.14 Mode modc
GB/T 3358.1-93
The point at which the density function or probability function reaches its maximum value. Note: If there is only one mode, the probability distribution is called unimodal. 2.15 Expectation
For a discrete random variable X that takes the value \, with probability p, its expectation is defined as: a.
where is the sum of all possible values ​​of X. Sp,r
For a continuous random variable X with density function f(r), its expectation is defined as: b.
The expectation of a random variable is also called the expectation of its probability distribution. Synonym: mean
Note: The above summation and integration require absolute convergence. Conditional expectationconditional expectation2.16
Expectation of the conditional distribution of a random variable.
Central random variablecentral random variable2.17
The difference between a random variable X and its expectation: XE(X). Note: The purpose of centralization is to make the expectation of the random variable zero. 2.18 Variancevariance
The variance of a random variable X is defined as:
f(r)da
V(X) = E[(X - E(X))27.
It is also called the variance of the distribution of X.
9 Standard deviationstandard deviation
The positive square root of the varianceV(X)
2.20 Coefficient of variationcocfficient of variationThe ratio of the absolute value of the standard deviation to the expectationVV(X)/IE(X)I. 2.21 Standardized random variable The quotient of the centralized random variable XE(X) and its standard deviation: X - E(X)
Note: The purpose of standardization is to make the expectation of the random variable 0 and the variance 1. 2.22 Moment
The expectation of the power function of the random variable. www.bzxz.net
2.23 moment about the origin The q-order moment about the origin of a random variable X (q is a positive integer) is the expectation of X raised to the fourth power: E(X\).
Note: The first-order moment about the origin is the expectation.
2.24 Central moment
The second-order central moment of a random variable X (g is a positive integer) is: Er(X - E(X))\].
Note: The second-order central moment is the variance,
2.25 Joint moment
GB/T 3358.1—93
The (4.s)-order joint origin moment (4+s are all positive integers) of the 3-dimensional random variable (X, Y) is: E(X'Y)
The (\s)-order joint central moment of the 3-dimensional random variable (XY) is: Er(X - E(x))\(Y -- E(Y))\..Note: The joint moment of multi-dimensional random variables can be defined similarly. 2.26 Covariance covariance
The covariance of a two-dimensional random variable (X, Y) is its (1, 1)-order joint central moment: E(Y)>)J.
Ca(X,Y) -- Fr(X -- E(X))(Y --2.27 Correlation coefficient correlation coeficicntThe correlation coefficient of a one-dimensional random variable (X, Y) is the ratio of its covariance to the product of the standard deviations of X and Y: Q
When (=0, the two random variables are said to be uncorrelated. 2.28 Absolute short momcnt
Ce(X,Y)
wv(x)v(y)
In the definitions of moments in clauses 2.23 to 2.25, quantities such as XX-E(X), Y, Y-E(Y) are replaced by their absolute values ​​IX/:X--E(X)I, IYI, IY-E(Y)I, etc., respectively. Moments defined in this way are called the corresponding absolute moments. Note: For absolute moments and, is an integer, but need not be an integer. Example 1: The qth order absolute central moment of X is:
E[IX- E(X))
Example 2: The (qs)th order absolute origin moment of (X, Y) is: E(IXYI).
2.29 Kurtosis
The kurtosis of the probability distribution of a random variable is the 4th order origin moment of the corresponding standardized random variable minus 3. It describes the steepness of the distribution of the standardized random variable.
Example: The kurtosis of a normal distribution is 0.
Skewness
The skewness of the probability distribution of a random variable is the third-order origin moment of the corresponding standardized random variable, which describes the degree of asymmetry of the distribution.
Note: When the skewness is , the distribution is called symmetric; when the skewness is not 0, the distribution is called skewed. When the skewness is positive, it is called right-skewed; when the skewness is negative, it is called left-skewed.
Example: The skewness of a normal distribution is 0.
2.31 ​​Regression function, regression equation and coefficient regression function, rcgression equaliou and rcgression couf.ficient
If X is a -dimensional random variable, then the conditional expectation \? when X=(\,,) is given, as a function of,, is called the regression function of . At this time, , is called an equation, and the graph of the regression function is called a regression surface. When the regression surface is a hyperplane y = b +h + b? +* +h*
, the regression is called linear. At this time, 6 is called the partial regression coefficient of Y to X (=1.2,,). When X is a -dimensional random variable, the graph of the partial regression function f) of Y to X is called the regression curve. When the regression line of Y to X is: ·The straight line y=αbr is called the regression line, and its slope h is called the regression coefficient of Y to x. 2.32 Uniform distribution uniform distribution a continuous probability distribution, its density function is 33
2.33 Normal distribution normal distribution GB/T 3358.1-93
f(α）
-a continuous probability distribution, its density function is f()=
代ux.0
Tip: This standard content only shows part of the intercepted content of the complete standard. If you need the complete standard, please go to the top to download the complete standard document for free.