WEBVTT
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Simulation is an important
tool in the analysis
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of probabilistic phenomena.
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For example, suppose
that X, Y, and Z
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are independent
random variables,
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and you're interested in
the statistical properties
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of this random variable.
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Perhaps you can find
the distribution
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of this random variable by
solving a derived distribution
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problem, but sometimes
this is impossible.
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And in such cases,
what you do is,
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you generate random samples
of these random variables
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drawn according to
their distributions,
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and then evaluate the function
g on that random sample.
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And this gives you one sample
value of this function.
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And you can repeat
that several times
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to obtain some kind of
histogram and from that,
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get some understanding about
the statistical properties
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of this function.
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So the question is, how can
we generate a random sample
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of a random variable whose
distribution is known?
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So what we want is to
create some kind of box
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that outputs numbers.
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And these numbers
are random variables
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that are distributed according
to a CDF that's given to us.
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How can we do it?
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Well, computers typically
have a random number generator
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in them.
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And random number generators,
typically what they do
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is generate values
that are drawn
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from a uniform distribution.
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So this gives us
a starting point.
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We can generate uniform
random variables.
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But what we want is to generate
values of a random variable
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according to some
other distribution.
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How are we going to do it?
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What we want to do is to create
some kind of box or function
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that takes this uniform random
variable and generates g of U.
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And we want to find the
right function to use.
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Find a g so that the
random variable, g of U,
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is distributed according to
the distribution that we want.
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That is, we want
the CDF of g of U
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to be the CDF
that's given to us.
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So let's see how we can do this.
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Let us look at the discrete
case first, which is easier.
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And let us look at an example.
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So suppose that I want
to generate samples
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of a discrete random variable
that has the following PMF.
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It takes this value
with probability 2/6,
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this value with probability
3/6, and this value
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with probability 1/6.
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What I have is a
uniform random variable
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that's drawn from a
uniform distribution.
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What can I do?
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I can do the following.
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Let this number here be 2/6.
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If my uniform random
variable falls
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in this range, which happens
with probability 2/6,
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I'm going to report
this value for
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my discrete random variable.
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Then I take an
interval of length 3/6,
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which takes me to 5/6.
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And if my uniform random
variable falls in this range,
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then I'm going to
report that value
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for my discrete random variable.
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And finally, with
probability 1/6,
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my uniform random variable
happens to fall in here.
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And then I report that [value].
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So clearly, the value
that I'm reporting
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has the correct probabilities.
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I'm going to report this
value with probability 2/6,
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I'm going to report that
value with probability 3/6,
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and so on.
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So this is how we can
generate random samples
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of a discrete
distribution, starting
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from a uniform random variable.
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Let us now look at what we did
in a somewhat different way.
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This is the x-axis.
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And let me plot the CDF of
my discrete random variable.
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So the CDF has a jump of 2/6, at
a point which is equal to that.
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Then it has another
jump of size 3/6, which
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takes us to 5/6 at
some other point.
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And that point here corresponds
to the location of that value.
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And finally, it has
another jump of 1/6
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that takes us to 1,
at another point, that
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corresponds to the third value.
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And look now at this
interval here from 0 to 1.
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And let us think as follows.
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We have a uniform
random variable
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distributed between 0 to 1.
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If my uniform random
variable happens
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to fall in this interval, I'm
going to report that value.
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If my uniform random
variable happens
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to fall in this interval, I'm
going to report that value.
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And finally, if my uniform
falls in this interval,
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I'm going to report that value.
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We're doing exactly the
same thing as before.
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With probability 2/6,
my uniform falls here.
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And we report this
value and so on.
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So what's a graphical
way of understanding
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of what we're doing?
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We're taking the CDF.
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We generate a value
of the uniform.
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And then we move
until we hit the CDF
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and report the
corresponding value of x.
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It turns out that
this recipe will also
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work in the continuous case.
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Let's see how this is done.
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So let's assume that
we have a CDF, which
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is strictly monotonic.
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So the picture
would be as follows.
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It's a CDF.
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CDFs are monotonic,
but here, we assume
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that it is strictly monotonic.
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And we also assume
that it is continuous.
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It doesn't have any jumps.
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So this CDF starts at 0 and
rises, asymptotically, to 1.
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What was the recipe that
we were just discussing?
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We generate a value for a
uniform random variable.
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We move until we hit the CDF,
and then report this value here
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for x.
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So what is it that we're doing?
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We're going from u's to x's.
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So we're using the
inverse function.
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The cumulative takes
as an input an x,
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a value on this axis, and then
reports, a value on that axis.
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The inverse function
is the function
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that goes the opposite way.
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We start from a value
on the vertical axis
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and takes us to the
horizontal axis.
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Now, the important thing is
that because of our assumption
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that f is continuous
and strictly monotonic,
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this inverse function
is well-defined.
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Given any point
here, we can always
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find one and only
one corresponding x.
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Now, what are the
properties of this method
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that we have been using?
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If I take some number c and then
take the corresponding number
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up here, which is
going to be F_X of c,
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then we have the
following property.
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My random variable
X is going to be
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less than or equal
to c if and only
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if my random variable X
falls into this interval.
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But that's equivalent to
saying that the uniform random
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variable fell in that interval.
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Values of the uniform
in this interval-- these
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are the values that
give me x's that are
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less than or equal to c.
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So the event that X is
less than or equal to c
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is identical to the
event that U is less than
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or equal to F_X of c.
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So this is how I am generating
my x's based on u's.
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We now need to verify that
the x's that I'm generating
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this way have the correct
property, have the correct CDF.
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So let's check it out.
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The probability that X is
less than or equal to c, this
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is the probability that U is
less than or equal to F_X of c.
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But U is a uniform
random variable.
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The probability of being
less than something
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is just that something.
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So we have verified that
with this way of constructing
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samples of X based
on samples of U,
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the random variable that
we get has the desired CDF.
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Let's look at an example now.
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Suppose that we want
to generate samples
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of a random variable, which is
an exponential random variable,
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with parameter 1.
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In this case, we
know what the CDF is.
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The CDF of an exponential with
parameter 1 is given by this
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formula, for non-negative x's.
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Now, let us find the
inverse function.
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If a u corresponds to 1
minus e to the minus x--
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so we started with some x here
and we find the corresponding
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u-- this is the formula that
takes us from x's to u's.
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Let's find the formula that
takes us from u's to x's.
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So we need to solve
this equation.
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Let's send u to the
other side, and let's
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send this term to
the left hand side.
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We obtain e to the minus
x equals 1 minus u.
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Let us take logarithms: minus
x equals to the logarithm
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of 1 minus u.
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And finally, x is equal to minus
the logarithm of 1 minus u.
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So this is the inverse function.
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And now, what we have
discussed leads us
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to the following procedure.
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I generate a random
variable, U, according
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to the uniform distribution.
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Then I form the
random variable X
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by taking the negative of
the logarithm of 1 minus U.
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And this gives me
a random variable,
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which has an exponential
distribution.
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And so we have found
a way of simulating
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exponential random variables,
starting with a random number
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generator that produces
uniform random variables.