### Some Applications of the Distribution of the Maximum on Independent Normal Random Variables

Roberto N. Padua

#### Abstract:

The paper deals with the distribution of the maximum of n independent normal random variables and some of its applications in the electricity power industry in the area of peak load estimation and in genetic selection for animal breeding. For small n, the difficulty in finding the value of multiple integrals involved in the distribution function of the maximum order statistics (and hence, in the computation of its expected value) is recognized. The paper provides for simple approximations to the mean of the largest order statistics both in the iid and non-identically distributed cases. Large sample asymptotic results for extreme values of normal random variables are often used in reliability theory and of late, used in the analysis of extreme weather changes in relation to climate change. While the large sample results for the iid case have been treated in the past, we focused on the relatively unexplored non-identical but independent case. Results show that : (a) the simple approximations to the mean of the largest order statistic both for the iid and non-iid cases have good MSE properties, and (b) the large sample distribution for a non-identically distributed case still obeys the Type I Gumbel distribution with shifted parameters.