Fundamentals of Applied Probability and Random Processes, 2nd Edition

7.5 Random Sum of Random Variables

Let X be a continuous random variable with PDF f_X(x) whose s-transform is M_X(s). We know that if Y is the sum of n independent and identically distributed random variables with the PDF f_X(x), then the s-transform of the PDF of Y is given by

${\mathit{M}}_{\mathit{Y}}\left(\mathit{s}\right)={\left[{\mathit{M}}_{\mathit{X}}\left(\mathit{s}\right)\right]}^{\mathit{n}}$

The above result assumes that n is a fixed number. However, there are certain situations when the number of random variables in a sum is itself a random variable. For this case, let N denote a discrete random variable with PMF p_N(n) whose z-transform is G_N(z). Our goal is to find the s-transform of the PDF of Y when the number of random variables is itself a random ...