Let $X_1,X_2,...$ be independent and identically distributed with uniform distribution over $(0,1)$ and $Y_1,Y_2,...$ be independent and identically distributed with density $e^{-x}\cdot\mathbb{I}\{x\ge0\}$.
How do I calculate
$$\lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge\sum^n_{i=1}Y_i\right)?$$
My instinct says that I should rewrite the expression so that I can use the weak law of large numbers but I don't know how.
Represent the limit probability equivalently as $$\lim_{n \to \infty} P\left(\frac{1}{n} \sum_{i=1}^n \left( X_i - Y_i \right) > 0 \right)$$
Then the average of $n$ samples sampled from the difference of these two distributions would converge to the expected value of the difference of these two distributions by the weak law of large numbers. In this case, the expected value would be equal to $\frac{1}{2}-1=-\frac{1}{2}$, so the limit would be equal to $0$ since $-\frac{1}{2} \not > 0$.