1. We would like to generate a Cauchy random variable X by starting from a uniformly dis- tributed random variable U Uniform[0; 1], and applying a suitable transformation g(:) mapping U to X so that X has the desired distribution. The Cauchy density function with parameter a is given by fX(x) = a= x2 + a2 : (a) Find the transformation g(:). (b) Write code (preferably in Matlab) to generate 10000 samples from a Cauchy distribution with a = 1 by applying the derived transformation to a uniformly distributed random variable and plot the data. You should be able to observe that the Cauchy distribution is heavy tailed. (c) Verify that the (empirical) distribution of the data matches the desired Cauchy distri- bution. 2. Let X and Y be statistically independent random variables with probability density functions pX(x) = 1 2 (x + 1) + 1 2 (x 1); pY (y) = 1 p 22 exp y2 22 and let Z = X + Y , and W = XY . (a) Find the conditional probability density functions pZ(zjx = 1) and pZ(zjx = 1). (b) Find the probability density function pZ(z) of Z. (c) Find the mean values mX = E[X];mY = E[Y ], the variances, 2Y ; 2W , and the covari- ance YW. Are Y and W uncorrelated random variables? Are Y and W statistically independent random variables? 3. A random variable X has probability distribution function PX(x) = [1 e2x]u(x) where u(:) is the unit-step function. (a) Calculate the following probabilities: P[X 1]; P[X 2]; P[X = 2]: (b) Find pX(x), the probability density function for X. (c) Let Y be a random variable obtained from X as follows: Y = 0 x < 2 1 if x 2 Find pY (y), the probability density function for Y . 4. Given the joint pdf fXY (x; y) of RVs X and Y . Find the pdf of Z = X2 +Y 2. You can write your answers in terms of the joint density fXY . (For your information: This type of problem arises in communication systems, for instance, when a signal is received with an unknown phase at a receiver. One strategy is to compute the power of the signal, which is the sum of squares of X and Y .)