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Numerical Methods for Financial Mathematics. Exercise Handout 3 Lecture: Prof. Dr. Christian Fries, Exercises: Dr. Lorenzo Torricelli, Tutorium: Panagiotis Christodoulou Sommersemester 2016 Exercise 1 Write a class which generates exponentially distributed random variables Xi of parameter λ = 0.2. Defining appropriate methods for this class, write a P program that uses both the Central limit theorem and Chebyshev’s inequality to the sample mean Y = ni=1 Xi /n with n = 1000 to find a 95% confidence interval for λ . Then repeatedly run the program and calculate the frequence with which λ is within the computed bounds. What are your conclusions? (you can use the inverseCumulativeProbability method in the class NormalDistribution of the org.apache.commons.math.distribution package.) Exercise 2 −1/3 over the interval [0, 8]. Does a Monte Carlo evaluation of the integral Consider R 8 the function f (x) = x I = 0 f (x)dx converge? If so, why? What if f (x) = x−1/2 ? And f (x) = 1/x? Let I˜ be the Monte Carlo integral estimator of I. Write a Java class with methods that evaluates I˜ for a given sample size, calculates the variance, and finds the standard error with the analytical value. Using this class, compute • Using M = 1000 drawings, compute the empirical probability of |I˜ − I| > 0.1 for various Monte Carlo sample sizes. Then find the minimum sample size N such that such a probability is roughly 5% and compare it with the value of N which can be extracted by the theoretical estimate found in the lectures script; • Write a method that returns a 99% confidence interval of I for any given sample size N . Exercise 3 Find D(Ai ) and D∗ (Ai ), i = 1, 2, for the sets: A1 = {1/8, 1/4, 1/2, 3/4} A2 = {1/4, 1/2, 5/8, 3/4}