Let $X_1, X_2, ..., X_n$ be independent random variables, uniformly distributed over the unit interval $[0, 1]$. Let $a$ be a real constant number satifying $\frac{n}{2} \leq a \leq n$, and denote $Y = \sum_{i = 1}^n X_i$. Estimate $P(Y \geq a)$ using the central limit theorem.
My attempt:
First observe that $X_1, X_2, ..., X_n$ have a common mean of $\frac{1}{2}$ and a common variance of $\frac{1}{12}$, let $Z = \frac{Y - \frac{n}{2}}{\sqrt{\frac{n}{12}}}$, we have $P(Y \geq a) = P(Z \geq \frac{a - \frac{n}{2}}{\sqrt{\frac{n}{12}}})$.
My question:
How should I do next to proceed? The normalized value is expressed in terms of $n$ and is it okay to express this inequality with function $\phi$ in terms of $n$? Also, how should I do to change from $a$ to $n$?
Yes, it is OK because convergence in distribution to a continuous distribution implies uniform convergence on the real line. Hence $P(Y \geq a)$ is approximated by $\int_{c_n} \phi (t) dt$ where $c_n=\frac {a-\frac n 2}{ \sqrt {n/12}}$