Bob wishes to check whether the total sum in his credit card statement is plausible. He adds up all figures concerning the purchases made, rounding each number to the nearest integer.
Let S be the cumulative error after summation. Suppose there were 100 purchases, and the fraction part of the cost of each purchase is equally likely to be any number from 0,0.01, ...,0.99. (The prices are often ending with 99c or 98c, but after taxation the distribution becomes more or less uniform.) If the fraction part is 0.5, Bob rounds to the nearest smaller or larger integer with equal probabilities. Find the mean and standard deviation of S.
What would you consider the accuracy of the approximation? Estimate the probability that the absolute value of the error will exceed $10; $15.