Problem Set 7 for Random Numbers

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Problem Set 7 for Random Numbers
Problem 1: As I warned (and showed) in class, there are many, many
deeply flawed pseudo-random number generators out there. One widely
found version is the default random number generator in the C standard
library. Look at test broken libc.py – this shows how to wrap the C standard
library in python, call its random number generator, and save the output.
(Note – the numba wrapper is there for speed but is not required.) I’ve
used it to generate random (x,y,z) positions with coordinates between 0 and
2
31 (the max random integer value in the standard library). The type of
PRNG used in the library is notorious for introducing correlations between
sequential points, with sets of points in n−dimensional space lying on a
surprisingly small number of planes.
To make this effect easier to see, I’ve pulled out all the (x,y,z) triples
with 0 < x, y, z < 108
(so about 5% of the total span) and put them in the
text file rand points.txt. Show that when correctly viewed, these triples lie
along a set of planes (I get about 30) and so are very much not randomly
distributed in 3D space. You can either do this by changing the view angle
on a 3D plot, or plotting ax + by, z for suitably chosen a and b. Do you
see the same thing happen with python’s random number generator? If
possible, can you see the same effect on your local machine? You may need
to change the name of the library in the line:
where libc.so would be standard under a Linux system, and you can google
for Windows. If you can’t get this part to work, that’s OK – just state so
and you won’t lose points.
Problem 2: We saw in class how to generate exponential deviates using
a transformation. Now write a rejection method to generate exponential
deviates from another distribution. Which of Lorentzians, Gaussians, and
power laws could you use for the bounding distribution? You can assume
the exponential deviates are non-negative (since you have to cut off the
distribution somewhere, might as well be at zero). Show that a histogram
of your deviates matches up with the expected exponential curve. How
efficient can you make this generator, in terms of the fraction of uniform
deviates that give rise to an exponential deviate?
Problem 3: Repeat problem 2, but now use a ratio-of-uniforms generator. If u goes from 0 to 1, what are your limits on v? How efficient is this
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generator, in terms of number of exponential deviates produced per uniform
deviate? Make sure to plot the histogram again and show it still produces