Introduction to Probability · Problem 9.3.2

Generating Functions — Grinstead & Snell

Mathematics — Probability · central-limit-theorem-continuous

Problem

Let

\{X_k\}

1 \leq k \leq n

, be a sequence of independent random variables, all with mean 0 and variance 1, and let

S_n

S^*_n

, and

A_n

be their sum, standardized sum, and average, respectively. Verify directly that

S^*_n = S_n/\sqrt{n} = \sqrt{n}A_n