There are many probability distributions that we did not cover in our inclass discussions. One such distribution, the Cauchy distribution, is used in spectroscopy and astronomy to describe the form of spectral lines subjected to homogeneous broadening. The Cauchy distribution has two parameters: x0 which describes the peak location, and describes the width of the distribution. The probability density function for the Cauchy distribution is given by: Help with question a and b please!

Figure 1: Example probability density functions for the Cauchy distribution. Image taken from Wikipedia. Recalling the precise definition of the mean of a distribution, =xp(x)dx, we can compute the mean of this distribution. The details of how the integral is evaluated are not important, but we can break up the integral into two parts and find, =0xp(x;x0,)dx+0xp(x;x0,)dx=+. (a) What is the mean of the Cauchy distribution? Rationalize and explain your answer. (b) Does the Central Limit Theorem apply when sampling from a Cauchy distribution? In other words, if we averaged n samples from a Cauchy distribution many times, would the resulting distribution of averages have a Normal distribution? Rationalize and explain your