A Recursive Formula for Moments of a Binomial Distribution

Abstract

While teaching a course in probability and statistics, one of the authors came across an apparently simple question about the computation of higher order moments of a random variable. The topic of moments of higher order is rarely emphasized when teaching a statistics course. The textbooks we came across in our classes, for example, treat this subject rather scarcely; see [3, pp. 265–267], [4, pp. 184–187], also [2, p. 206]. Most of the examples given in these books stop at the second moment, which of course suffices if one is only interested in finding, say, the dispersion (or variance) of a random variable X , D2(X) = M2(X)− M(X)2. Nevertheless, moments of order higher than 2 are relevant in many classical statistical tests when one assumes conditions of normality. These assumptions may be checked by examining the skewness or kurtosis of a probability distribution function. The skewness, or the first shape parameter, corresponds to the the third moment about the mean. It describes the symmetry of the tails of a probability distribution. The kurtosis, also known as the second shape parameter, corresponds to the fourth moment about the mean and measures the relative peakedness or flatness of a distribution. Significant skewness or kurtosis indicates that the data is not normal. However, we arrived at higher order moments unintentionally. The discussion that follows is the outcome of a question raised by our students without knowing much about the motivation for studying moments of higher order. It has to do with discrete random variables