2.4 Find the transformation to obtain, from an observation U following a continuous uniform (0, 1) distribution, an observation from each of the following continuous probability distributions:

(a) Exponential distribution with mean 1.

(b) Beta distribution with a¼2 and b¼1. The probability density function is given by f (x) ¼ 2x for 0 < x < 1

(c) The logistic distribution defined by the probability density function f (x) ¼
e(xa)=b [1 þ e(xa)=b]2 1 < x < 1, 1 < a < 1, 0 < b < 1

(d) The double exponential distribution defined by the probability density function f (x) ¼
1 2b e(jxaj)=b 1< x < 1, 1< a < 1, 0 < b < 1

(e) The Cauchy distribution defined by the probability density function f (x) ¼
b p[b2 þ (x  a)2] 1 < x < 1, 1 < a < 1, 0 < b < 1