Chapter 5, Problem 76SE

In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information about the total cost distribution by adding together characteristics of the individual component cost distributions—this is called the “roll-up” procedure. For example, E(X₁ +…+ Xn) = E(X₁) 1…..+ E(X_n), so the roll-up procedure is valid for mean cost. Suppose that there are two component tasks and that X₁ and X₂ are independent, normally distributed random variables. Is the roll-up procedure valid for the 75th percentile? That is, is the 75th percentile of the distribution of X₁ + X₂ the same as the sum of the 75th percentiles of the two individual distributions? If not, what is the relationship between the percentile of the sum and the sum of percentiles? For what percentiles is the roll-up procedure valid in this case?