# Math Help - Strong law of large number

1. ## Strong law of large number

Suppose let $(X_n)_{n\ge 1}$ be i.i.d. with N(1,3) random variables. How can I show that:

$\displaystyle\lim_{n\to\infty}\frac{X_1+...+X_n}{X _1^2+...+X_n^2}=\frac{1}{4}$ (a.s)

From my understanding: $\displaystyle\lim_{n\to\infty}\frac{S_n}{n}=\displ aystyle\lim_{n\to\infty}\frac{1}{n}\displaystyle\s um_{j=1}^{n}X_j=\mu$
Then the nominator will go to 1.. How about the denominator? Why is its expected value 4?

2. Hello,

It's not exactly the numerator that will go to 1, but it's $X_1+\dots+X_n/n$

You have $\displaystyle \frac{X_1+\dots+X_n}{X_1^2+\dots+X_n^2}=\frac{X_1+ \dots+X_n}{n}\cdot\frac{n}{X_1^2+\dots+X_n^2}$

For the second term, we know - also from the SLLN, that it tends to $\mathbb{E}[X_1^2]$. But we know that $3=Var[X_1]=\mathbb{E}[X_1^2]-(\mathbb{E}[X_1])^2$, so the result should be...