Mathematicians came up with these distributions through the use of convolution. As a quick introduction, if there are two independent and identically distributed (i.i.d.) random variables, X and Y, and where their respectively known probability density functions (pdf) are f<\/i>x<\/sub>(x) and f<\/i>y<\/sub>(y), we can then generate a new probability distribution by combining X and Y using basic summation,multiplication, and division. Some examples are listed above, e.g., the F distribution is a division of two Chi\u2010Square distributions, the normal distribution is a sum of multiple uniform distributions, etc.To illustrate how this works, consider the cumulative distribution function (cdf) of a joint probability distribution between the two random variables X and Y:<\/p>\n Differentiating the cdf equation above yields the pdf:<\/p>\n Example 1: The convolution of the simple sum of two identical and independent uniform distributions approaches the triangular distribution.<\/p>\n As a simple example, if we take the sum of two i.i.d. uniform distributions with a minimum of 0 and maximum of 1, we have:<\/p>\n Where for a Uniform [0, 1] distribution, f (x) =1 when 0 <= x <=1, we have:<\/p>\n Which approaches a simple triangular distribution.<\/p>\n The figure below shows an empirical approach where two Uniform [0, 1] distributions are simulated for 20,000 trials and their sums added. The computed empirical sums are then extracted and the raw data fitted using the Kolmogorov\u2010Smirnov fitting algorithm in Risk Simulator. The triangular distribution appears as the best\u2010fitting distribution with a 74% goodness of fit. As seen in the convolution of only two uniform distributions, the result is a simple triangular distribution in Risk Simulator. The triangular distribution appears as the best\u2010fitting distribution with a 74% goodness of fit. As seen in the convolution of only two uniform distributions, the result is a simple triangular distribution.<\/p>\n Example 2: The convolution simple sum of twelve identical and independent uniform distributions approaches the normal distribution.<\/p>\n If we take the same approach and simulate 12 i.i.d. Uniform [0, 1] distributions and summed them, we would obtain a very close to perfect Normal distribution as shown below, with a goodness of fit at 99.3% after running 20,000 simulation trials.<\/p>\n Example 3: The convolution simple sum of multiple identical and independent exponential distributions approaches the gamma (Erlang) distribution.<\/p>\n In this example, we sum two i.i.d. exponential distributions and generalize it to multiple distributions.To get started, we use two identical Exponential [\u03bb = 2] distributions:<\/p>\n where f (x)= \u03bbe-\u03bbX<\/sup> is the pdf for the exponential distribution for all x>=0;\u03bb>=0 and the distribution\u2019s mean is \u0392=1\/\u03bb.<\/p>\n If we generalize to n random i.i.d. exponential distributions and apply mathematical induction:<\/p>\n This is, of course, the generalized gamma distribution with \u03b1 and \u03b2 for the shape and scale parameters:<\/p>\n When the \u03b2 parameter is a positive integer, the gamma distribution is called the Erlang distribution, used to predict waiting times in queuing systems, where the Erlang distribution is the sum of independent and identically distributed random variables each having a memoryless exponential distribution. Setting n as the number of these random variables, the mathematical construct of the Erlang distribution is:<\/p>\n The empirical approach is shown below where we have two exponential distributions with \u03bb = 2 (this means that the mean \u03b2 = 1\/\u03bb = 0.5). The sum of these two distributions, after running 20,000 Monte Carlo simulation trials and extracting and fitting the raw simulated sum data, shows a 99.4% goodness of fit when fitted to the gamma distribution where the \u03b1 = 2 and \u03b2 = 0.5 (rounded), corresponding to \u00a0 n = 2 and \u03bb = 2.<\/p>\n The standard definition of copulas is based on Sklar\u2019s Theorem, which states that an m\u2010dimensional copula (or m\u2010copula) is a function C from the unit m\u2010cube [0, 1]m to the unit interval [0, 1] that satisfies the following conditions:<\/p>\n Consider a continuous m\u2010variate distribution function F ( y1<\/sub>\u2026.ym<\/sub> ) with univariate marginal Distributions F1<\/sub>(y1<\/sub>)\u2026 Fm<\/sub>(ym<\/sub>) and inverse quantile functions F1<\/sub> -1<\/sup>\u2026. Fm -1<\/sup>. Then we have y1<\/sub>=F1<\/sub>-1<\/sup>(u1<\/sub>)-F1<\/sub>,,,,,,,ym<\/sub>=Fm<\/sub>-1<\/sup>(um<\/sub>)–Fm<\/sub> where u1<\/sub>…..um<\/sub> are uniformly distributed variates. Therefore, the transforms of uniform variates are distributed as Fi<\/sub>(i=1,,,,m). This means we have:<\/p>\n where C is the unique copula associated with the distribution function. That is, y ~ F , and F is continuous, then F1<\/sub>(y1<\/sub>),,,,,Fm<\/sub>(ym<\/sub>)~C, and if U ~ C , then we have F1<\/sub>-1<\/sup>(u1<\/sub>),,,,,Fm<\/sub>-1<\/sup>(um<\/sub>) ~ F. \u00a0Mathematical algorithms using Iman\u2010Conover and Cholesky decomposition matrices are used to compute the joint marginal distributions. Copulas are parametrically specified joint distributions generated from given marginals. Therefore, properties of copulas are analogous to properties of joint distributions.<\/p>\n Convolution theory is applicable and elegant for theoretical constructs of probability distributions. With basic addition, multiplication, and division of known i.i.d. distributions, we can determine its theoretical outputs. The issue with convolution theory is that there are no correlations (independently distributed) between the random variables and their distributions, and the individual distributions have to be exactly the same (identically distributed) and commonly known.<\/p>\n Therefore, if one modifies the distributions, uses exotic distributions, mixes and matches different non\u2013i.i.d. distributions, adds correlations, and creates large Excel models (beyond the simple addition, multiplication, or division as shown above, such as when there are exotic financial models and computations), truncation, empirical nonparametric distributions, historical simulation, and other combinations of such issues, convolution will not work and cannot predict the outcomes. In addition, both convolution and copula theorems can only be used to compute correlations of joint distributions but would be limited to only a few distributions before the mathematics become intractable due to the large matrix inversions, multiple integrals and differential equations that need to be solved. Therefore, users are restricted to using Monte Carlo risk simulations.<\/p>\n![\"CONVOLUTION](\"http:\/\/rovdownloads.com\/blog\/wp-content\/uploads\/2015\/04\/CONVOLUTION-AND-COPULAS.jpg\")
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