Copyright c
±
2010 by Karl Sigman
1
Inverse Transform Method
Assuming our computer can hand us, upon demand, iid copies of rvs that are uniformly dis
tributed on (0
,
1), it is imperative that we be able to use these uniforms to generate rvs of any
desired distribution (exponential, Bernoulli etc.). The ﬁrst general method that we present is
called the inverse transform method.
Let
F
(
x
)
, x
∈
IR
,
denote any cumulative distribution function (cdf) (continuous or not).
Recall that
F
: IR
→
[0
,
1] is thus a nonnegative and nondecreasing (monotone) function that
is continuous from the right and has left hand limits, with values in [0
,
1]; moreover
F
(
∞
) = 1
and
F
(
∞
) = 0. Our objective is to generate (simulate) rvs
X
distributed as
F
; that is, we
want to simulate a rv
X
such that
P
(
X
≤
x
) =
F
(
x
)
, x
∈
IR
.
Deﬁne the generalized inverse of
F
,
F

1
: [0
,
1]
→
IR, via
(1)
F

1
(
y
) = min
{
x
:
F
(
x
)
≥
y
}
, y
∈
[0
,
1]
.
If
F
is continuous, then
F
is invertible (since it is thus continuous and strictly increasing)
in which case
F

1
(
y
) = min
{
x
:
F
(
x
) =
y
}
, the ordinary inverse function and thus
F
(
F

1
(
y
)) =
y
and
F

1
(
F
(
x
)) =
x
. In general it holds that
F

1
(
F
(
x
))
≤
x
and
F
(
F

1
(
y
))
≥
y
.
F

1
(
y
) is a nondecreasing (monotone) function in
y
.
This simple fact yields a simple method for simulating a rv
X
distributed as
F
:
Proposition 1.1 (The Inverse Transform Method)
Let
F
(
x
)
, x
∈
IR
,
denote any cumu
lative distribution function (cdf) (continuous or not). Let
F

1
(
y
)
, y
∈
[0
,
1]
denote the inverse
function deﬁned in (1). Deﬁne
X
=
F

1
(
U
)
, where
U
has the continuous uniform distribution
over the interval