‘lb receive credit, you must do your own work, using the notation and methods Write up your answers neatly and clearly, including a description of what you are doing along with mathematical derivations and computer programs. Note: parts 1 and 2 are to be done analytically, while part 3 involves writing a program and generating output. You may use Matlab or Python for the computer program.
1. Let Fx(x) be the cdf of a continuous random variable X . Assume that Fx (x) is strictly increasing. Then the inverse FX1(u) exists (on it E (0, 1)) and is strictly increasing. Let U be a uniform random variable on (0,1) and let V = Fx-1(U). (a) Sketch a typical Fx (x) and the corresponding Fx-1 (u) . (b) Prove that {u E (0,1) : Fx-1 (u) < v} = {u E (0,1) : u < Fx (v)} . (c) Prove using part (b) that Fir (v) = Fx (v) .
(Hence V has the same cdf and pdf as X, and can be generated from a uniform random variable U which is generated by a standard random number generator).
2. The Laplacian random variable has a pdf 1 fx (x) = V2o-2 exp -V2/021 xl for – oo < x < co Show that v = Fx-1(u) has the following form
v = .072/2 ln(2u), for for 0 < u < 1/2 1/2 < u < 1 vci2/21n(2(11-u))’
3. Let cr2 = 1. Write a program to generate 10000 realizations of a uniform random variable on (0,1), and use the formula derived in part 2 to convert them into 10000 realizations of a Laplacian random variable. Plot both the pdf of the Laplacian random variable and a normalized histogram of these 10000 samples of the Laplacian random variable .
