What is the chi-square distribution? Meaning and uses

　・In Japanese
Prerequisites
　・Probability Distribution
　・Standard normal distribution
　・Gamma function

■What is the chi-square distribution?

Chi-square distribution is the probability distribution of χ (chi) expressed as the sum of squares as shown below when n random variables Z₁, Z₂, ... Zn that follow a standard normal distribution with a mean of 0 and a variance of 1 are sampled. In this case, the number of samples n is called the degrees of freedom. Also, Γ is called the gamma function.



The graph below shows this.

■Meaning of the chi-square distribution

For example, when a single sample that follows a standard normal distribution is taken (when the degree of freedom is n=1), the most likely value that the sample can take is 0. In other words, χ_n=0 is the most likely, so f(χ_n) at that time will be the largest.

Also, when three samples are taken (when the degrees of freedom n = 3), the samples will not all take values ​​close to 0, so we can imagine that χ_n will be greater than 0. So which value is most likely? This means that the value around 3 is the most likely.

■Applications of the chi-square distribution

By measuring χ_n when multiple samples are taken, it is possible to check whether the sample really follows the standard normal distribution. If χ_n is significantly different from the theoretical value, it is clear that the sample is an out-of-standard part. This is called the chi-square test.

List of related articles TOP MENU Probability / Statistics Statistics/analysis Expected Value Variance, Covariance standard deviation Root Mean Square least squares formula Weighted/Moving average Gaussian process regression Kernel function Basis functions Ridge/Lasso regression Sequential Bayesian Estimation Probability Joint probability Conditional probability Weibull distribution Chi-square distribution