The Moments of log _e ( S ) From a Nonstandard Normal Universe

ABSTRACT

This article is a sequel to the previous by Maghsoodloo and Silva, where the authors derived the sampling distribution of log _e ( S ) from the normal universe N( μ , 1). Our first objective is to generalize to the sampling distribution of y  = log _e ( S ), where the variate S represents the standard deviation of a random sample of size n ≥ 2 from a normal universe of any standard deviation σ > 0. We will then obtain the exact expressions for the first 4 moments of Ln( S | σ > 0), and the tabulations of the exact first 4 moments are provided in Tables 1 and 2; Table 1 values are then verified by Taylor's approximation; Table 3 compares the exact values of E[log _e ( S )] versus those from the Taylor expansion.