Let $X_1, X_2, \dots, X_{1200} \to U(-1/2,1/2)$. Find $P\{S>20\}$ and $P\{|S| > 20\}.$

Question

Exercise :

Let $X_1, X_2, \dots, X_{1200}$ be independent and uniformly distributed in the interval $(-1/2,1/2)$.

If : $S=X_1 + X_2 + \dots + X_{1200}$, estimate the probabilities $P\{S>20\}$ and $P\{|S| > 20\}$.

Sorry for not providing an attempt but I'm at a loss on how to start here. I would really appreciate a thorough explanation.

All I can say is that we can observe that :

$$X_1, X_2, \dots, X_{1200} \to U(-1/2,1/2)$$

which means that :

$$f_{X_1}(x_1)=f_{X_2}(x_2)=\dots=f_{X_{1200}}(x_{1200})= 1$$

Please, assist me with this problem as it's an exam question that I am trying to grasp for tomorrow's semester exams.

Answer 1

From the Central limit theorem:

$$ S_{1200} =\sum^{1200}_{i=1}X_i \sim N(n\mu,n\sigma^2) $$

here $S_{1200} \sim N(0, 100)$