This says the probability that a random variable with our pdf is less than one is 25%
. This
gives rise to a clever observation that underlies many methods to generate non-uniform
random numbers. We want a function f()
that when we call it as f(drand48())
we get a return
value with a pdf ¼ x^2
. We don’t know what that is, but we do know that 25% of what it returns
should be less than 1, and 75% should be above one. If f()
is increasing, then we would expect
f(0.25) = 1.0.
This can be generalized to figure out f()
for every possible input:
f(P(x)) = x
That means f
just undoes whatever P
does. So,
f(x) = P
-1
(x)
The -1 means “inverse function”. Ugly notation, but standard. For our purposes what this
means is, if we have pdf p()
and its cumulative distribution function P()
, then if we do this to a
random number we’ll get what we want:
e = P
-1
(drand48())
For our pdf p(x) = x/2
, and corresponding P(x)
, we need to compute the inverse of P
. If we
have
y = ¼ x^2
we get the inverse by solving for x
in terms of y
:
x = sqrt(4y)
Thus our random number with density p we get by:
e = sqrt(4*drand48())
Note that does range 0 to 2 as hoped, and if we send in ¼ for drand48()
we get 1 as desired.
We can now sample our old integral
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