Chi-Squared_Distribution-卡方分布
# Chi-Squared Distribution
2021-12-04
Tags: #Math/Statistics
A very Good Website Chi-squared distribution - Wikipedia
# 重要结论
$Z_{1},Z_{2},\cdots,Z_{n}$ are independent standard normal random variables,
- The Sum of independent Standard Normal Variables are still Normal: $$X=Z_{1}+Z_{2}+\cdots+Z_{n}$$
Link: 正态分布_高斯分布_Normal_Distribution-Gaussian_Distribution
- 但是正态独立随机变量的平方和是卡方分布的:
$$Y=Z_{1}^{2}+Z_{2}^{2}+\cdots+Z_{n}^{2}$$ then $Y$ is said to have a chi-squared distribution with $n$ degrees of freedom shown by $$Y \sim \chi^{2}(n)$$
卡方分布其实是一种特殊的Gamma分布: $$Y\sim Gamma\left(\frac n2,\frac1 2\right)$$
下面是卡方分布的图像随着自由度的变化:
