Been studying statistics lately. Been reading about the t-distribution and how as the degree's of freedom approaches infinity this distribution approaches the standard normal distribution. Can somebody give me an intuitive explanation of why this is?
Also, if you want to talk about any other relation between these two relations, I'd appreciate it. Thanks!
If you know the Central Limit Theorem, you know that the sum of iid random variables converges to the normal distribution. The t-distribution is the distribution of $T_n:=\frac{\bar{X_n}-\mu}{S_n/ \sqrt{n}}$ where $\bar{X_n}$ is the sample mean of $n$ iid Gaussian random variables and $S_n$ the sample standard deviation. We define the parameter $\nu:=n-1$. As $\nu \to \infty$ we know that this size of the sample being modeled is getting very large and so $T_n \to N(0,1)$ by The normality of the sample mean and and noting that $S_n/ \sqrt{n} \to 1$ in probability
A related “fun fact” of the t distribution is that it converges to the standard Cauchy distribution as degrees of freedom approach 1. William Gossett’s discovery is really a remarkable unification of inference from normally distributed populations.
