Statistics 简明教程

Statistics - Chi-squared Distribution

自由度为 k 的卡方分布（卡方分布或 ${X^2}$ - 分布）是 k 个独立标准正态随机变量平方和的分布。它是统计学中使用最广泛的概率分布之一。它是伽马分布的一种特殊情况。

The chi-squared distribution (chi-square or ${X^2}$ - distribution) with degrees of freedom, k is the distribution of a sum of the squares of k independent standard normal random variables. It is one of the most widely used probability distributions in statistics. It is a special case of the gamma distribution.

卡方分布被统计学家广泛用于计算以下内容：

Chi-squared distribution is widely used by statisticians to compute the following:

Estimation of Confidence interval for a population standard deviation of a normal distribution using a sample standard deviation.
To check independence of two criteria of classification of multiple qualitative variables.
To check the relationships between categorical variables.
To study the sample variance where the underlying distribution is normal.
To test deviations of differences between expected and observed frequencies.
To conduct a The chi-square test (a goodness of fit test).

Probability density function

卡方分布的概率密度函数给出为：

Probability density function of Chi-Square distribution is given as:

Formula

其中——

Where −

${\Gamma(\frac{k}{2})}$ = Gamma function having closed form values for integer parameter k.
${x}$ = random variable.
${k}$ = integer parameter.

Cumulative distribution function

卡方分布的累积分布函数给出为：

Cumulative distribution function of Chi-Square distribution is given as:

Formula

其中——

Where −

${\gamma(s,t)}$ = lower incomplete gamma function.
${P(s,t)}$ = regularized gamma function.
${x}$ = random variable.
${k}$ = integer parameter.