On Normal Approximations of the Frequency Functions of Standard Forms Where the Main Variables are Normally Distributed

Risk analysts frequently encounter functional relationships concerning costs and revenue which can be expressed as standard algebraic forms. The forms are quadratic, product, or mixed functions of variables which are normally distributed and not necessarily statistically independent. (The ratio functions will hopefully be the subject of a paper that follows this.) Generally, interest focuses on the construction of 2\sigma or 3\sigma probability intervals for each form. Since probability distributions of these forms, as far as the risk analyst is concerned, are neither tabled nor easily derived, the desired probability intervals are not easily constructed. This paper shows through analysis, review of previous research, and simulation, that under ordinary conditions the probability distribution of these forms can be approximated by more population distributions (e.g., gamma or normal). The paper suggests rules of thumb that allow one to make probability statements about each form without recourse to costly simulation. An application to a financial planning model for an industrial firm augments the theoretical discussion.