Hi,

I am confused by the role of Laplace transform and Foureir transform in
probability theory and their relations.

Fourier transform of a PDF is called the characteristic function.
Laplace transform of a PDF is called a probability generating function.

1. Fourier transform of a PDF always exists. How about the Laplace
transform?

2. We can always do inverse fourier transfrom to obtain the PDF, but
one PDF have both a Laplace transform and a Fourier transform, so the
Laplace transform of the very PDF ought to be invertible? Is it always
invertible?

3. Since the two transforms corresponding to the same PDF, in order for
both of them to work, the singularities of the Laplace-transformed
function must be all to the left of the imaginary axis, thus the
inverse Laplace transform's integral path should be on the right to the
imaginary axis, so that the inverse Laplace transform can be reduced to
Fourier transform always?

4. Using characteristic function, we often meet with nested
characteristic functions, for example, random variable X has
characterisc function f(u), Y has characteristic function g(v), X and Y
have a compounded relation that their transform functions become
nested, say the overall transform function is f(g(v)), but u is
supposed to be real in the original definition of the characteristic
function, now g(v) is a complex number, so it has a problem here. Can
the definition of characteristic function and probability generating
function, and moment generating function all be expanded from real to
complex?

Please help me clarify my confusions... thx

In article .com,
Lucy wrote:
Hi,

I am confused by the role of Laplace transform and Foureir transform in
probability theory and their relations.

Fourier transform of a PDF is called the characteristic function.
Laplace transform of a PDF is called a probability generating function.

This is not the case. A probability generating function
is used for distributions on the integers, and while it
CAN be related to the Laplace transform, this is rarely
of much use.

1. Fourier transform of a PDF always exists. How about the Laplace
transform?

It always exists for distributions on [0, infinity). For
other distributions, with a sign change it is the moment
generating function.

2. We can always do inverse fourier transfrom to obtain the PDF, but
one PDF have both a Laplace transform and a Fourier transform, so the
Laplace transform of the very PDF ought to be invertible? Is it always
invertible?

One method of inverting to get either the density or
the CDF is to integrate on the appropriate line in the
complex plane with fixed real part.


3. Since the two transforms corresponding to the same PDF, in order for
both of them to work, the singularities of the Laplace-transformed
function must be all to the left of the imaginary axis, thus the
inverse Laplace transform's integral path should be on the right to the
imaginary axis, so that the inverse Laplace transform can be reduced to
Fourier transform always?

See the above. There ARE real variable methods for inverting
the Laplace transform (or the moment generating function), but
they can only be used reasonably in closed form expressions,
not numerically without fantastic precision.

4. Using characteristic function, we often meet with nested
characteristic functions, for example, random variable X has
characterisc function f(u), Y has characteristic function g(v), X and Y
have a compounded relation that their transform functions become
nested, say the overall transform function is f(g(v)), but u is
supposed to be real in the original definition of the characteristic
function, now g(v) is a complex number, so it has a problem here. Can
the definition of characteristic function and probability generating
function, and moment generating function all be expanded from real to
complex?

I find this paragraph to be confusing. Can you give an
example to show what you mean?

Please help me clarify my confusions... thx



--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

Herman Rubin wrote:
In article .com,
Lucy wrote:
Hi,

I am confused by the role of Laplace transform and Foureir transform in
probability theory and their relations.

Fourier transform of a PDF is called the characteristic function.
Laplace transform of a PDF is called a probability generating function.

This is not the case. A probability generating function
is used for distributions on the integers, and while it
CAN be related to the Laplace transform, this is rarely
of much use.

1. Fourier transform of a PDF always exists. How about the Laplace
transform?

It always exists for distributions on [0, infinity). For
other distributions, with a sign change it is the moment
generating function.

2. We can always do inverse fourier transfrom to obtain the PDF, but
one PDF have both a Laplace transform and a Fourier transform, so the
Laplace transform of the very PDF ought to be invertible? Is it always
invertible?

One method of inverting to get either the density or
the CDF is to integrate on the appropriate line in the
complex plane with fixed real part.


3. Since the two transforms corresponding to the same PDF, in order for
both of them to work, the singularities of the Laplace-transformed
function must be all to the left of the imaginary axis, thus the
inverse Laplace transform's integral path should be on the right to the
imaginary axis, so that the inverse Laplace transform can be reduced to
Fourier transform always?

See the above. There ARE real variable methods for inverting
the Laplace transform (or the moment generating function), but
they can only be used reasonably in closed form expressions,
not numerically without fantastic precision.

4. Using characteristic function, we often meet with nested
characteristic functions, for example, random variable X has
characterisc function f(u), Y has characteristic function g(v), X and Y
have a compounded relation that their transform functions become
nested, say the overall transform function is f(g(v)), but u is
supposed to be real in the original definition of the characteristic
function, now g(v) is a complex number, so it has a problem here. Can
the definition of characteristic function and probability generating
function, and moment generating function all be expanded from real to
complex?

I find this paragraph to be confusing. Can you give an
example to show what you mean?


HI Herman,

I am digesting your other answers and will get back to you later. For
question 4, one example is:

Let Y=sum(X_i, i from 1 to N), where X_i's are iid, and N is a positive
integer-valued random variable. Let's call X_i's characteristic
function g(v), and N's characteristic function f(v).

then if I can treat log(g(v)) / i (where "i" is an imaginary unit) as a
whole thing, and substitute it as "v" into f(v), then I will have the
overall characteristic function:

f( log(g(v)) / i )

---------------------

I am wondering if this is allowed because normally "v" has to be a real
number, from the definition of characteristic function...