Let $X_1, X_2...$ be independent and identically distributed with mean 4 and variance 20. Set $S_n = X_1 + X_2 + ... + X_n$ and $V_n = X_1^2 + X_2^2 + … + X_n^2$. Use the Strong Law of Large Numbers to find a constant $c$ such that $P\left(\lim_{n \to \infty} \frac{S_n}{V_n} = c\right) = 1$. I've tried to find the expected value of $\frac{S_n}{V_n}$ as I believe that should be the answer, which I keep computing, incorrectly, as $\frac{1}{104}$. What should I be doing here instead?
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Note that $$ \frac{S_{n}}{V_n}=\frac{S_n/n}{V_{n}/n}\to\frac{EX_{1}}{EX_{1}^2} $$ a.s. as $n\to \infty$ by the SLLN and limit laws. You should able to compute $EX_{1}^2$ from the givens.
answered Apr 28, 2020 at 21:21