Normal Distribution
By the central limit theorem, any variable that is the sum of a large number of independent factors is likely to be normally distributed.1 $X\sim N(\mu,{\sigma}^{2})$ denotes normal distribution with mean $\mu$ and variance${\sigma}^{2}$
Standard Normal Distribution
variable mean is 0, and standard deviation is 1. $X\sim N(0,1)$ denotes X meets standard normal distribution.
Chi-square Distribution
distribution of the sum of square of n independent standard normal random variables is called chi-square distribution with n degree of freedom and is written as ${\chi}^{2}_{n}$. Density function of $\chi$ only depends on degree of freedom n.
Suppose $Z\sim N(0,1)$ and $U\sim {\chi}^{2}_{n}$ with Z and U independent, $t=\frac{Z}{\sqrt{U/n}}$, the distribution t is the t-distribution with n degree of freedom.
ratio of two independent chi-squares. Let $U\sim {\chi}^{2}_{m}$ and $V\sim {\chi}^{2}_{n}$, $F=(U/m)\div(V/n)$ is called F-distribution with m and n d.f., written as $F\sim {F}_{m,n}$.
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