The relationship between the three major sampling distributions and the normal distribution

Abstract

The chi-square distribution, tdistribution and Fdistribution are the three most important sampling distributions in probability theory and mathematical statistics. They all have a certain relationship with the normal distribution, and they are asymptotically normal under certain conditions. This article first combines the gamma distribution and the moment generating function indicate the correlation properties of the three major sampling distributions and the relationship between the three. Using the relevant functions such as the characteristic function and the central limit theorem to explain that they all converge to the normal distribution under certain conditions. At the same time, they use mathematical software MATLAB to verify.