CSI 779 Assignment 1

1.  The Weibull distribution has density function,

      p(x) = a/b  x^(a-1) exp(- x^a / b), for x > 0.

    How would you generate random variates from this distribution
    using the inverse CDF method?

2.  Give an algorithm to generate a normal random deviate using
    the acceptance/rejection method with the double exponential
    density as the majorizing density.  
    The double exponential has density p(x) = t/2 exp (|tx|).
    After you have obtained the acceptance/rejection test, try
    to simplify it.

3.  Obtain a Monte Carlo estimate of the base of the natural
    logarithm.  Give 95% confidence bounds for your estimate.

4.  Use crude Monte Carlo to estimate the value of the integral
      \int _0 ^\infty x^2 sin(\pi x) exp(-x/2) dx.

5.  Repeat problem 4. using antithetic random variables.

6.  Generate a pseudorandom sample of size 100 from a N(0,1) 
    distribution that has a sample mean of 0 and a variance of 1.

7.  Devise a "dice test" for a random number generator.
    Apply it to the runif generator in S-Plus.

8.  Write a program to generate d-dimensional Halton sequences,
    using the integers j, j+1, j+2, ..., and using the first
    d primes as the bases.

9.  Apply the dice test you developed in problem 7. to the output
    of the program for