MY MATEX NOTES

          In a previous post we showed how useful Fourier Transform may be in computing convolution of symmetric R.V.s. The Laplace tranform plays an analogous role for the convolution of one sided non-negative R.V.s. In this post We will show it´s usefulness in handling with the convolution of Levy-Smirnov R.V.s., an alpha stable distribution. Laplace transform: Recall the Laplace transform pair (1) (2) Now recall  from the previous post the convolution of two non-negative R.V.s If We take the it´s Laplace transform We obtain If and are i.i.d we obtain Similarly for the sum of three R.V.s Taking it´s Laplace transform Keeping on this process We may obtain (3) By equation (2) We Obtain (4) Levy-Smirnov distribution:      The Levy-Smirnov distribution is a continuous probability distribution for a non-negative random variable. It belongs to the family of alpha-stable distributions. Like all stable distributions ...

      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...