Compute $\lim_{n\to\infty} \frac{\sum_{i=1}^{n} X_{i}}{\sum_{i=1}^{n}X_{i}^2 }$ where $X_i$'s are i.i.d uniform on $(0,1)$

Question

Let $X_{i}$ be i.i.d. Uniform(0,1) random variables. Compute

$$\lim_{n\to\infty} \frac{\sum_{i=1}^{n} X_{i}}{\sum_{i=1}^{n}X_{i}^2 }$$

Do we use the expected value of both to solve this?

Edit :I am thinking of using Wald's equation $E[X_{1} + X_{2} + ..... X_{n} ]= E[N]*E[X_{1}]$

And what to do about the square term in the denominator.

Answer 1

Just divide the numerator and the denominator by $n$ and use SLLN to conclude that the almost sure limit is $\frac {EX_1} {EX_1^{2}}=\frac 3 2$. [Note that $(X_i^{2})$ is also an i.i.d. sequence].