My tax return involves 32 different numbers, each rounded to the nearest dollar and then added together. Assuming that the errors by rounding are uniformly distributed on the interval (-1/2,1/2), estimate the probability that the sum of the rounded amounts differs from the true sum by less than $1. Hint: this is really a question about the sum of the rounding errors.
This is a homework problem and I have no idea where to even begin. Any help would be greatly appreciated.
A random variable uniformly distributed on $\left(-\frac12,\frac12\right)$ has mean $0$ and variance $\frac1{12}$
so the sum of $32$ independent random variables with the same distribution has mean $0$ and variance $\frac{32}{12}=\frac83$. The value $1$ would then be $\sqrt{\frac38}$ standard deviations above the mean, and $-1$ the same below
Using a normal approximation (justified using a central limit theorem argument) would suggest that the probability of the sum being in the interval $(-1,1)$ could be about $\Phi\left(\sqrt{\frac38}\right)-\Phi\left(-\sqrt{\frac38}\right)$ where $\Phi\left(x\right)$ is the cumulative distribution function of a standard normal random variable with mean $0$ and variance $1$. So about $0.45971$
With a lot of effort involving numerical convolution, it is possible to do a more precise calculation, and I did so in 2008 in a note called May not sum to total due to rounding: the probability of rounding errors. On page 16 it gave a value for $G_{32}(1)$ of about $0.45804$
The sum can be approximated by a nomrally distributed error. Its mean is obviously $0$, the variance is the sum of the variances of the single errors.
