The main purpose of this thesis is to apply an algorithm for the numerical inversion of the Laplace transform that recovers the probability density function (PDF) of a sum of nonnegative continuous random variables. The Laplace transform is used in many disciplines. For example, in actuarial sciences, a common application is to study the distribution of the sum of nonnegative independent random variables. Because it is a popular method, numerical techniques have been developed to invert the Laplace transform. In the discrete case, by using a moment generating function (MGF) of a sum of independent discrete variables, the distribution can be analytically determined. In the continuous case, if the MGF fails to determine the distribution of a sum of nonnegative continuous independent variables analytically, then the PDF of the sum will be recovered by numerically inverting the Laplace transform.