1. Show that the t-distribution with r = 1 degree of freedom and the Cauchy distribution are the same.
2. Let F have an F-distribution with 2r and 2s degrees of freedom. Since the support of F is (0, ∞), the F-distribution is often used to model time until failure (lifetime). In this case, Y = log F is used to model the log of lifetime. The log F family is a rich family of distributions consisting of left- and right-skewed distributions as well as symmetric distributions; see, for example, Chapter 4 of Hettmansperger and McKean (2011). In this exercise, consider the subfamily where Y = log F and F has 2 and 2s degrees of freedom.
(a) Obtain the pdf and cdf of 
.
(b) Using R, obtain a page of plots of these distributions for s = .4, .6, 1.0, 2.0, 4.0, 8. Comment on the shape of each pdf.
(c) For s = 1, this distribution is called the logistic distribution. Show that the pdf is symmetric about 0.