Description
1. Use a number generator to produce samples of
a) 10
b) 100
c) 1000
d) 10000
standard uniformly distributed random variables. For every sample size, produce a plot
that shows the empirical cumulatice distrbution function Fˆ in comparison with the
actual cumulative distibution function F.
2. Use a number generator to produce samples of
a) 10
b) 100
c) 1000
d) 10000
standard uniformly distributed random variables. For every sample size, produce a
histogram that shows the relative frequency of values taken, rounding to one decimal
place (i.e., you count the relative frequency of realizations in bins of size 0.1).
2
3. A good way to check if a sequence of random variables is close to being indeed random
is to make a correlation plot: Plot the sequence of pairs (xn, xn+1) generated by the
random number generator inside the unit square. As less structure is visible in the plot,
as better the random number generator is. Make correlation plots for a sequence of
pseudorandom variables for
a) The built-in uniform random number generator with seed ˜x0 = 375;
b) A linear congruence random number generator with m = 11, a = 6 and c = 0 (and
seed ˜x0 = 1);
c) A linear congruence random number generator with m = 231 − 1, a = 16807 and
c = 0 (and seed ˜x0 = 1);
d) A linear congruence random number generator with m = 231 − 1, a = 950706376
and c = 0 (and seed ˜x0 = 1).
4. The generation of random number generators before the Mersenne Twister improved
upon the basic linear congruence generator by coming different random number
generators. E.g., The Wichmann-Hill generator implemented in Python before version
2.3 sums up over different LCRNGs and takes the fractional part of the sum.
Specifically, assuming that there are K random number generators, working for
k ∈ {1, . . . , K} by
x0,k = ˜x0,k
xi+1,k = akxi,k mod mk
ui,k =
xi,k
mk
one calculates
ui =
X
K
k=1
ui,k −
$X
K
k=1
ui,k%
(where bxc denotes the largest integer smaller or equal than x).
Consider the specific case of two LCRNGs with ˜x0,1 = 3, a1 = 5, m1 = 7 and
x˜0,2 = 1, a2 = 7, m2 = 5.
a) Calculate the period length of the two LCRNGs as well as the combined
Wichmann-Hill generator.
b) Make plots for the serial correlation of all three generators (as in problem 3).
Note: All programming problems should be either in Python 2.7 (recommended) or
Python 3.5, matlab, or R (no support for these languages provided). Please comment
the programs extensively and send them in a .zip file with title Lastname HW1.zip and
suject line ”MA 573 HW1 Lastname” to Qingyun Ren qren@wpi.edu before the due
date of the homework (replacing the bold words by your actual last name). Plots can be
provided either as printout or as .pdf file.
6 points per problems
