Download Exercise 1 Write a class which generates exponentially distributed

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}