541 Homework #5 (due 30 November 2006) PROBLEM 3 UPDATED NOVEMBER 28


1) give an example of a Linear Congruential random number generator that does
  not have "full period", e.g., the modulus is m, but there exist integers
  over [1,m-1] which the generator may not create.  For this
    - identify a, c, and m
    - work out an example which shows that from some starting seed x,
      the generator does not have full period.

2) Given the ability to generate a stream of independent uniform u(0,1)
random variables, describe how to generate a random permutation
of the integers 1 through N, where any of the N! possible permutations is equally
likely.

3) Consider a non-homogeneous Poisson process, with rate function
  L(t) = exp{-5t} + 0.1,  and let X be the second event to occur
  during any sample path.  Write a computer program that generates
  1000 instances of X, and estimates the mean and constructs a 95\% confidence
  interval for it.  Turn in the program, a textual description of what it does,
  a description of how you generated the confidence interval, with the
  mean and confidence interval.  Be sure to describe in detail your method for generating
  random numbers, and random variates.