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,

,

are independently distributed as

Obtain the approximate distribution of the total error

in the sum of the n numbers and hence find the probability that the absolute error in the sum is at most

.

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

~

From the Uniform dist, I know E(X) = 0, and Var(X) =

So does this mean that the approximate dist of

is

~

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