The sum of random variables should not be confused with the sum of their distributions. If both distributions are normal, the former is also a normal distribution with appropriately scaled parameters. The latter would be what is called a Gaussian mixture. This piece will illustrate both sums, beginning with two normal distributions with identical standard deviations yet different means. First, we load necessary packages and define our distributions: pacman::p_load("ggplot2", "RColorBrewer", "extrafont") set.

Cash may be king, but cashflow is god—at least for small businesses, who can be simultaneously profitable yet bankrupt. A creditor, perhaps with cash flow problems of its own, need only delay the payment of its accounts receivable long enough such that the small business in question lacks the resources to meet its short-term obligations. Unfortunately, even fractional CFOs may be disinterested in a small business due to its size. The responsibility of cash flow management therefore falls frequently on the shoulders of those who feel ill equipped to manage what amounts to the differencing of statistical distributions over time.