“chi-square distribution”的概念、定义、翻译、参考文献-科学参考

chi-square distribution

Physics

The distribution of the sum of the squares of random variables with standard normal distributions. For example, if x₁, x₂,…, x_i,…are independent variables with standard normal distribution, then

$χ^{2} = \sum x_{i}^{2}$

has a chi-square distribution with n degrees of freedom, written χ‎_n^r. The mean and variance are n and 2n, respectively. The values χ‎_n²(α‎) for which

$P (χ^{2} \leq χ_{n}^{2} (α)) = α$

are tabulated for various values of n.

The chi-square test is a measure of how well a theoretical probability distribution fits a set of data.

Economics

A continuous distribution, denoted χ‎²_(n), described by the density function

$f (x) = \frac{1}{Γ (n / 2) 2^{n / 2}} x^{\frac{n}{2} - 1} e^{- x / 2}, 0 \leq x < \infty, n = 1, 2 \dots .$

and the moment generating function

$M (t) = {(1 - 2 t)}^{- n / 2}, t < \frac{1}{2}$

The parameter n is termed ‘the degrees of freedom’.

It is a particular case of gamma distribution (α‎ = n/2, β‎ = 2). This distribution can be derived from the normal distribution and arises in econometrics as the distribution of sums of certain random variables. In particular, if z is a standard normal random variable then z² has χ‎²₍₁₎-distribution, and if z₁,…,z_n are independent standard normal random variables then the sum z₁² +…+ z_n² has χ‎²_(n)-distribution. A χ‎²_(n)-random variable has mean n and variance 2n. It is the limiting distribution of the scaled F-distribution and is used for hypothesis testing in many econometric applications.

单词	chi-square distribution
释义	chi-square distribution Physics The distribution of the sum of the squares of random variables with standard normal distributions. For example, if x₁, x₂,…, x_i,…are independent variables with standard normal distribution, then $χ^{2} = \sum x_{i}^{2}$ has a chi-square distribution with n degrees of freedom, written χ‎_n^r. The mean and variance are n and 2n, respectively. The values χ‎_n²(α‎) for which $P (χ^{2} \leq χ_{n}^{2} (α)) = α$ are tabulated for various values of n. The chi-square test is a measure of how well a theoretical probability distribution fits a set of data. Economics A continuous distribution, denoted χ‎²_(n), described by the density function $f (x) = \frac{1}{Γ (n / 2) 2^{n / 2}} x^{\frac{n}{2} - 1} e^{- x / 2}, 0 \leq x < \infty, n = 1, 2 \dots .$ and the moment generating function $M (t) = {(1 - 2 t)}^{- n / 2}, t < \frac{1}{2}$ The parameter n is termed ‘the degrees of freedom’. It is a particular case of gamma distribution (α‎ = n/2, β‎ = 2). This distribution can be derived from the normal distribution and arises in econometrics as the distribution of sums of certain random variables. In particular, if z is a standard normal random variable then z² has χ‎²₍₁₎-distribution, and if z₁,…,z_n are independent standard normal random variables then the sum z₁² +…+ z_n² has χ‎²_(n)-distribution. A χ‎²_(n)-random variable has mean n and variance 2n. It is the limiting distribution of the scaled F-distribution and is used for hypothesis testing in many econometric applications.
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