Determination of the distribution function of a random variable that is the sum of several random variables

An algorithm for calculating integrals with a parameter when calculating differential and cumulative distribution functions of a random variable that is the sum of several random variables with known distribution densities has been developed. The results of testing the proposed methods at relatively small values of the second derivative of the integrand expression have shown high efficiency of using the numerical method of mean rectangles with a constant value of the partition interval. The error of calculations at the number of intervals $n=100\div1000$ in this case is about $10^{-4}$. Increasing $n$ to $10^4\div10^5$ reduces the error to $10^{-5}\div10^{-15}$. If the second derivative of the integrand is large or is determined by the distribution density for the sum of more than two random variables, more accurate solutions are required. Among them, we note the use of variable interval in combination, for example, with Simpson's or Gauss's methods, which provide higher accuracy.