Could someone please provide some explanation to the following question ?
Let $X_{1},X_{2},...$ be independent random variables that are uniform in [0,1]; and let $S_{n} = X_{1} + ... + X_{n}$. Using the CLT, give an approximation of $P(S_{100} \subset [45,55])$.
The problem with me is that I know how to solve such problems in discrete cases (ie: flipping a coin or tossing a dice) but how would I apply this case since I believe that there can be a large number of values that $X_{1}$ can take on (correct me if I'm wrong).
So I am a little bit stuck and would greatly appreciate it if someone could provide some help.
Thank you
Outline:
The sum $S$ of $100$ iid $Unif(0,1)$ random variables will be very nearly normal.
Also, $E(S) = \mu = 100E(U) = 100(1/2) = 50$ and by independence $V(S) = \sigma^2 = 100V(U) = 100(1/12).$
So a reasonable approximation is $P\{(45-\mu)/\sigma < Z < (55-\mu)/\sigma\},$ where $Z$ is standard normal.