# [SciPy-User] [OT] Transform (i.e., Fourier, Laplace, etc.) methods in Prob. & Stats.

David Goldsmith d.l.goldsmith@gmail....
Wed Nov 25 17:24:37 CST 2009

Good info, thanks; I'll look up "your" thread, Josef, on the archive and run
down what look like relevant references.  (FWIW, my interest is that I'm
helping out (nominally, "tutoring," but this level, it's more akin to being
a sounding board, checking his derivations, and "reminding" him of various
subtleties that are emphasized in math, but not necessarily in EE, etc.)
this guy working on his dissertation on air traffic control automation using
wireless communication protocols, very probability heavy stuff, and for the
second time yesterday, he presented me with a transform application - in
this instance, the "Z" transform - in this probability-heavy stuff, and this
is outside of my training in probability, so I want to "bone-up.")  Thanks
again,

DG

On Wed, Nov 25, 2009 at 2:41 PM, nicky van foreest <vanforeest@gmail.com>wrote:

> Hi,
>
> 2009/11/25  <josef.pktd@gmail.com>:
> > On Wed, Nov 25, 2009 at 2:53 PM, David Goldsmith
> > <d.l.goldsmith@gmail.com> wrote:
> >> Are there enough applications of transform methods (by which I mean,
> >> Fourier, Laplace, Z, etc.) in probability & statistics for this to be
> >> considered its own specialty therein?  Any text recommendations on it
> (even
> >> if it's only a chapter dedicated to it)?  Thanks,
> >>
> >
> > Some information is in the thread on my recent question
> > "characteristic functions of probability distributions"
> >
> > There is a large literature in econometrics and statistics about using
> > the characteristic function for estimation and testing.
> > The reference of Nicky for queuing theory uses mostly the Laplace
> > transform (for discrete distributions),
>
> It has been some time ago (more than 5 years...), but I recall that
> Whitt, in his articles on the numerical inversion of Laplace
> transforms, discretized Laplace transforms to facilitate the
> inversion, The distributions themselves are not necessarily discrete.
> One example would be the waiting time distribution of customers in a
> queue, which is continuous for most service and arrival processes.
>
> There is certainly potential for dedicated numerical inversion algo's
> for the Laplace transforms of density and distribution functions. The
> latter form a somewhat specialized sort of function. Distribution
> functions are 0 at -\infty, and 1 at \infty, and are non decreasing.
> They may also have discontinuities, but not too many. These properties
> may affect the inversion.  Besides these properties, the transforms
> are used to obtain insight into the behavior of the sum of independent
> random variables. Such sums can be rewritten as the product of the
> transforms of distribution. This product in turn requires inversion
> to, as some people call it, take away the Laplacian curtain.
>
> Nicky
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