MY MATEX NOTES

      Today´s post will be more on applied math. We will discuss about sum of Random variables and convolution. First we will find the expression of the density function of a sum of two Random Variables and then apply it to find the density of the sum of two exponential R.V.´s and then to the sum of n exponential R.V.´s which ends up by being a Gamma R.V. which is important for the Poisson process and Renewal theory. We then recall the fourier transform and use it to find the density function of the sum of n Gaussian R.V.´s and n Cauchy R.V.´s. Interesting fact is that the sum of n Gaussians has a gaussian distribution and the sum of n Cauchy´s has a cauchy distribution, due to an propertie called stability. Convolution of two independent random variables Let , where X and Y are two independent and identical distributed continuous R.V. Given , what is ? We know that , so if we know we can find . Then or We can then conclude that (1)      An...

                In probability theory and Stochastic processes the Characteristic function or Fourier transform of a random variable plays an important role, specially in problems involving summations of independent random variables. It´s much easier to compute analytically these sums in the Fourier space. As with the distribution function and the density function, the characteristic function characterizes completely the r.v. Additionally, some r.v´s don´t have an expression for their density given by elementary functions, and we express them by their characteristic functions (most stable distributions). In this post I want to compute the Fourier transform of two stable distributions, namely the Gaussian and the Cauchy distributions. These two, aside from being the only two symmetric stable distributions with a density given by elementary functions, they also represent THE EXAMPLES of a Thin tailed distribution and a Fat tail...