Originally Posted by

**Zenter** I'm given a question which asks the following:

In adding n real numbers, each is rounded to the nearest integer. Assume that the round-off errors, $\displaystyle X_{i}$, $\displaystyle i = 1,...,n,$ are independently distributed as $\displaystyle U(-0.5,+0.5)$

Obtain the approximate distribution of the total error $\displaystyle \sum X_{i}$ in the sum of the n numbers and hence find the probability that the absolute error in the sum is at most $\displaystyle \frac{1}{2}\sqrt{n}$.

Now, I assume it wants me to approx to Normal dist by CLT, which is:

$\displaystyle

\frac{\sqrt{n}(\overline{X} - \mu)}{\sigma}$ ~ $\displaystyle N(0,1)$

From the Uniform dist, I know E(X) = 0, and Var(X) = $\displaystyle \frac{1}{12}$

So does this mean that the approximate dist of $\displaystyle \sum X_{i}$ is

$\displaystyle \overline{X_{i}}\sqrt{12n}$ ~ $\displaystyle N(0,1)$

or am I missing something regarding the fact that it's the dist of the sum rather than just X itself?