Approximating the Distribution for Sums of Products of Normal Variables

Abstract We consider how to calculate the probability that the sum of the product of variables assessed with a Normal distribution is negative. The analysis is motivated by a specific problem in electrical engineering. To resolve the problem, two distinct steps are required. First, we consider ways in which we can assess the distribution for the product of two Normally distributed variables. Three different methods are compared: a numerical methods approximation, which involves implementing a numerical integration procedure on MATLAB, a Monte Carlo construction and an approximation to the analytic result using the Normal distribution. The second step considers how to assess the distribution for the sum of the products of two Normally distributed variables by applying the Convolution Formula. Finally, the two steps are combined to compute the distribution for the sum of products of Normally distributed variables, and thus to calculate the probability that this sum of products is negative. The problem is also approached directly, using a Monte Carlo approximation.