1. Pseudorandom Numbers
-by Alex Roodman

2. Part 1: What does
“Pseudorandom” mean?
The word “Pseudorandom”
can be split into these two parts:
pseudo, random. The “random” part is
self explanatory, but then there is
“pseudo.”
“Pseudo” implies that it only
mimicks randomness. This is often bad for
random numbers, but some computer
programs require random numbers, such
as programs for reaction tests. You don’t
want someone to be able to predict what
they’re reacting to, and the given output
will still be generated by a computer.
Computers can’t do better that generate
pseudorandom numbers.
This presentation is about
different simple pseudorandom
number generators, and their
qualities. The results will be
presented via. histogram.
One quality of a
pseudorandom number generator is
that it is equally likely to be any
number in a specified range. If you
flip a coin several times, it’s equally
likely to be either heads or tails.
Another quality is how random
the changes in the numbers are. If you
count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, the
number you count at any given moment
is equally likely to be any number from 1
to 10, but it’s still not random.

3. Part 2: How the Difference
works
Understanding the
first quality discussed in this
presentation is trivial: The
numbers need to be generated
about as often as each other.
The difference works differently,
though.
Imagine this: You are
rolling a fair die. First, you roll a 1.
The next number rolled is equally
likely to be any number from 1 to 6.
Thus, the difference between 1 and
the next number rolled is equally
likely to be 0, 1, 2, 3, 4, or 5. We
can make a small histogram of this,
repeated. For 2, the number can
change by -1, 0, 1, 2, 3, or 4. For 3,
we have -2, -1, 0, 1, 2, and 3, etc.
The histogram would look
like this:
Now we know
what to expect in the
difference of the
pseudorandom numbers.
-5
-4
-3
-2
-1
0
1
2
3
4
5

4. Generator #1
𝑔1 𝑥 ≡ e 𝑥
mod 1
If you take 𝑎 𝑥mod 𝑏,
(where both 𝑎 and 𝑏 are integers,)
and count 𝑥, then as long as 𝑎 and 𝑏
share no factors, you can get back
each number from 1 to 𝑏 − 1
exactly once before you get the
same number as before. Thus,
modular exponentiation is uniformly
distributed.
In simpler
words, if you multiply
a number by itself
several times, and
subtract another
number from it as
many times as
possible without going
below 0, you will get a
random number. This
is the method I tested
here.
Based on this, the
pseudorandom numbers
generated here will be well
distributed. The only problem that
could happen is if there is a bad
distribution of the differences of
numbers.

5. Testing
Generator #1
As you see below, the
distribution of the numbers only has a
few bumps in it. The distribution of the
differences is also in the triangle shape
that we wanted.
The only disappointment
is that the generator can’t generate
many numbers. If you do exactly
what the generator says, you can
only generate 31 numbers. If you
use a simpler way of calculating the
numbers, you can generate up to
713 numbers. This is a good
pseudorandom number generator,
but the calculations are just too
much for the computer.
= Difference
= Numbers Generated

6. Generator #2
𝑔2 𝑥 = sin2 𝑥
This function is
basically a wave that goes up
and down very quickly. One way
to represent this is taking a
spring, stretching it, and
flattening it sideways. You will
see a wave.
If you look at the
wave, you will see that it
stays at either the far top or
the far bottom for a bit
before it continues. This
could cause a problem with
the distribution.

7. Testing Generator #2
As expected,
the numbers are
generated more often
on the sides.
What’s curious,
though, is why the difference
in the numbers is always
around 0. This means that
the number generated
slowly travels back and forth,
between 1 and 0.
= Difference
= Numbers Generated

8. Generator #3
Imagine that you spin
a spinner, and it keeps spinning
forever. It keeps going at the
same speed all the time. It
takes 10 seconds to make a full
rotation, and there are 10
numbers total. Every 7
seconds, you write down what
number the spinner is over.
This is equally likely to be any
number on the spinner. We use
this concept in this generator.
The only thing that
might go wrong is the distribution
of the differences.
𝑔3 𝑥 ≡ 𝑒𝑥 mod 1

9. Testing Generator #3
The distribution of the
numbers generated is near perfect.
The distribution of the differences,
though, is a different matter. The difference between
the outputs can be exactly one of
two numbers. This is a little
disappointing, because we were
sort of relying on that for this to be
a good generator. The last generator
wasn’t that good, but at least it
could have a difference of more
than two numbers.
= Difference
= Numbers Generated

10. Generator #4
𝑔4 𝑥 ≡ ln 𝑥 mod 1 This generator is (spoiler alert) the
worst yet. It is intended, though, because using
this, you can see what bad qualities you don’t
want to see in a pseudorandom number
generator. The difference between the numbers
in this generator get smaller and smaller, and it
doesn’t even have a distribution.
Mathematicians, before you call this bogus, go
to
https://www.khanacademy.org/cs/pseudorand
om-number-generators/4733195911168000,
and look for something that says //Change to
try different generators. Look to the left of it,
and change the number in the brackets to 4.
Click restart, and watch.

11. Testing Generator #4
As expected, this wasn’t a very
good generator. The distribution of the
differences is basically a single spike,
The
distribution here, if you
keep generating
numbers, won’t ever
stay the same. By this,
we can’t tell whether
the distribution here is
even or spiky.
= Difference
= Numbers Generated

12. Conclusion
It seems that most of these generators
are terrible. The best one of them is too tough on
the computer, even though it works beautifully.
This shows how hard it is to
create these generators. One of the best
generators used is called the Mersenne
Twister. This generator is hard to
understand, and involves the
distribution of Mersenne Primes.
Making these good generators is
extremely difficult.