Click for printer friendely version of this HowTo Sampling From a Distribution Suppose you want to do some experiments that explore the stochastic nature of a process. For example, you incorporate a random variable into a model. The first thing you must do is create some observations that have variation according to the distribution of the random variable. Most often, the method used to do this is to invert the Cumulative Distribution Function (described below) and feeding it a random number between 0 and 1.3.3 It is possible to convert from a uniform(0,1) random variable to something nonuniform, and also the reverse. Although this may sound backward, and in fact is, we will start by describing how to do the reverse. Figure 3.8.1 shows a typical exponential distribution, which starts at 1 and ends near zero. The Cumulative Distribution Function (or CDF) for the same distribution can be used to map an exponential random variable to a uniform(0,1) variable and is shown in Figure 3.8.2. The CDF for any distribution is defined as the cumulative probability from the smalled number in the domain to a specific point in the distribution, $p$. That is, $P(X \le p)$. If $X$ is distributed by the exponential function $f(x) = e^{-x}$, then $$P(X \le p) = \int_{0}^{p} e^{-x} \textrm{d}x$$ $$= \left[-e^{-x} \vert^{p}_{0} \right]$$ $$= 1 - e^p.$$ Since the total area under any distribution is always 1, that is for any distribution, $g(x)$, $$\int_{-\infty}^{\infty} g(x) \textrm{d}x = 1$$ it is clear that the CDF for any distribution can be used to map any random variable from that distribution to a number between 0 and 1. Since no part in the CDF will be visited any more than any other part, the mapping is to a uniform(0,1) distribution. Figure: PDF of an exponential process with rate = 1: $pdf(x) = e^{-x}$ Figure: CDF of an exponential process with rate = 1: $cdf(x) = 1 - e^{-x}$ To map a uniform(0,1) random variable to an exponential distribution you simply do the reverse. That is, use the inverse of the exponential distribution CDF, $x = -\log(1-y)$, and apply it to a uniform(0,1) variable. Here is some Octave code that demonstrates this transformation (see Figure 3.8.3 for the histogram): octave:1> uniforms = rand(50, 1); octave:2> exps = -log( 1 - uniforms); octave:3> hist(exps) # plot histogram Figure: A histogram of 50 exp(1) random variables This method, in general works very well for the subset of distributions that are formed by transforming uniform random variables. These include the Beta, Gamma, Chi-squared, F, Exponential, Double Exponential and Weibull distributions. Other distributions rely on tricks to generate random variables. Despite the fact that this text attempts to avoid describing tricks at all costs, due to their utility we will list two of them. Click for printer friendely version of this HowTo

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