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

Figure 16

histogram: cross markers (black color), numerical simulation; circle markers (red color), analytical solution. The PDF of is a chi distribution with two degrees of freedom, which is shifted by 400 unit, and the one of is a uniform distribution. and are 400 and 441, respectively. 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. ; ; .