Modified Normal-based Approximation to the Percentiles of Linear Combination of Independent Random Variables with Applications

A modified normal-based approximation for calculating the percentiles of a linear combination of independent random variables is proposed. This approximation is applicable in situations where expectations and percentiles of the individual random variables can be readily obtained. The merits of the approximation are evaluated for the chi-square and beta distributions using Monte Carlo simulation. An approximation to the percentiles of the ratio of two independent random variables is also given. Solutions based on the approximations are given for some classical problems such as interval estimation of the normal coefficient of variation, survival probability, the difference between or the ratio of two binomial proportions, and for some other problems. Furthermore, approximation to the percentiles of a doubly noncentral F distribution is also given. For all the problems considered, the approximation provides simple satisfactory solutions. Two examples are given to show applications of the approximation.