The Roads from Normal Distrubution to F Distribution

Here is a pathway we could connect:

• Normal Distribution
• Chi-squared Distribution
• F-Distribution
• Student’s t-test

1. If $Z_i,i=1,2,3,\dots k$ are independently distributed,

   Then we could say: ${\sum }_{i=1}^k{Z}_i^2\sim {\chi }^2(k)$

2. If $Z\sim N(0,1)$ and $V\sim {\chi }^2(v)$, then $T_k=\frac{Z}{\sqrt{\frac{C_k}{k}}}\sim t(k)$

The reason why we have to study different kinds of distributions is that we have to deal with different random samples. Different distributions can fix different situations. Such as Normal Distribution is good at dealing the issues when the sample is relatively large, and t-distribution, F distribution, and Chi-squared Distribution could deal with the situation when sample is small.

It’s hard to say which tool is better, we have to focus on which problem we are trying to solve.