The sum of independent random variables (SIRV), � � �where �has been discussed by many authors particularly on the moments and the distributions. It is assumed that �are independent random variables, and N is a discrete random variable and independent of �The sum of independent random variables also called the random sum has been applied on queuing theory, risk theory, population models and biometrics, otherwise, some authors have discussed the sum of correlated random variables. The sum of correlated random variables can arise in insurance,��finance, sports, cellular mobile systems, internet traffic, and communications. This paper discusses calculation of moments (expectation and variance) of the sum of correlated random variables, where N is a discrete random variable and independent of �Some assumptions on moments of N and X need to be set to calculate the moments. A simple numerical verification uses to verify the formulas that derived in this paper.