Abstract
We present a streamlined proof of a formula for the derivatives of the moment generating function of the multivariate normal distribution. We formulate it in terms of the summation of the contractions by pairings, which encodes a combinatorial computation procedure. We give two applications. First, we provide a simple proof of Isserlis’ theorem and derive a formula for the moments of the multivariate normal distribution. Second, we calculate the moments of the product of a finite number of correlated normally and lognormally distributed random variables.
Reference10 articles.
1. T. Alberts and D. Khoshnevisan, Calculus on Gauss Space: An Introduction to Gaussian Analysis., 2018, book in preparation available at http://www.math.utah.edu/ davar/math7880/F18/GaussianAnalysis.pdf.
2. A hybrid multivariate Normal and lognormal distribution for data assimilation
3. K. Hirose, Topics in quantitative analysis of social protection systems., 1999, Issues in Social Protection, Discussion Paper No. 6. International Labour Office, Social Security Department, Geneva.
4. ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES
5. Continuous Multivariate Distributions