An Estimate of the Probability Density Function of the Sum of a Random Number of Independent Random Variables

Figure 1

histogram: cross markers (black color), numerical simulation; circle markers (red color), analytical solution. The PDF of is formed by three delta of Dirac, with , , and ; ; the PDF of is a chi square distribution with two degrees of freedom. The null hypothesis is the normal PDF with mean and variance given by (3) and (4), respectively. The means and standard deviations of and are ; ; ; . The sample number is 10000 and the number of bins is 60. The chi squared test is accepted at the significance level of 0.05. ; ; .