R Notebook

Chapter 9 Problem 2

Let $ X_k, 1 k n $ be a sequeunce of independent random variables, all with mean 0 and variance 1. Let \(S_n, S_n^*\), and \(A_n\) be their sum, standardized sum, and average, respectively. Below, we verify that \[ S_n^* = \frac{S_n}{\sqrt n} = \sqrt n A_n \implies \\ S_n = n A_n = S_n^*\sqrt n \implies \\ A_n = \frac{S_n}{n} \] Which is true by the above definition.