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2.4.3 Cauchy Distribution

This is an interesting pdf, because strictly speaking, its mean does not exist and its variance is infinite. Given our definition of mean,

we find that this integral does not exist because the separate integrals for x > 0 and x < 0 do not exist. However, if we allow a ``principal value'' integration, where the limits are taken simultaneously, we see that the integral for x < 0 will cancel the integral for x > 0 and the mean is zero, consistent with a graphical interpretation of this pdf, as depicted in Figure 8. However, if we try to compute the variance, we find:

which is an unbounded integral. Thus if we sample from the Cauchy distribution and we attempt to predict the extent to which samples will fall ``close'' to the mean, we will fail. Note that the Cauchy distribution is a legitimate pdf, because it satisfies the properties of a pdf given in Eq. (24) and Eq. (25), namely,

but its variance is infinite and its mean necessitates a more general definition of integration.

Figure 8: Cauchy Probability Distribution Function.

These have been examples of single random variable, or univariate, pdf's. Let us now consider bivariate pdf's, which generalize readily to multivariate pdf's (the important conceptual step is in going from one to two random variables). Bivariate distributions are needed for a number of important topics in Monte Carlo, including sampling from multidimensional pdf's and the analysis of rejection sampling.