probability help?

Question

how do you know to use the uniform distribution and why is the distribution of U(-0.5,0.5) ?

In a laboratory, 48 measurements are taken and rounded off
to the nearest integer. Find the probability that the total error
introduced by rounding is no more than 2.

Answer 1

The total error is the sum of the 48 individual errors due to rounding.

If the measurements are uniformly distributed, then the distance of any one measurement from the true value to the round-off value is going to be uniformly distributed over (-0.5, +0.5) (If you were truncating instead of rounding, the error would be uniformly distributed over (0, 1) instead.)

So the total error is the sum of 48 independent uniformly distributed random variables.

The Central Limit Theorem says that for all practical purposes, the sum is going to be normally distributed:
http://en.wikipedia.org/wiki/Central_limit_theorem

To use the approximation, check out:
http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29

and

http://en.wikipedia.org/wiki/Normal_distribution

Note, however, that you are right to question the assumption of uniformity. In the real world, measurements are generally not uniformly distributed.

http://en.wikipedia.org/wiki/Benford's_law

Still, school problems are usually over-simplifications.