Could be, but "X/Y is Cauchy", Statistical Inference, Casella and Berger,
p626.
----- Original Message -----
From: Peter M Lee <[log in to unmask]>
To: <[log in to unmask]>
Sent: Tuesday, February 17, 2004 11:02 AM
Subject: Re: distribution of 1/X , if X belongs to a normal distribution
> I think you're confusing this with the correct result that (X+Y)/(X-Y) has
> a Cauchy distribution - Peter M Lee
>
> --
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> ][_n_i_|( |_ | | | Wentworth College | | pml1
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> ( _ | ||| | Peter M Lee | | University of York | | @york.ac.uk
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> On Tue, 17 Feb 2004, Jeppe H. Rich wrote:
>
> > >From somewhere deep down in my memory, I have the idea that if X and Y
> > are normal then Z=X/Y will be Cauchy distributed. Please do not take
> > this for granted but check out. Of course, as Phillip points out you
> > should be careful with the domains here.
> >
> > A second thing, why not use log-normal distributions in your simulation?
> > They are invariant to inversion.
> >
> > /Jep
> >
> > ----------------------------------
> > Jeppe Husted Rich, MSc, PhD
> > Assistant Professor
> > Technical University of Denmark
> > Centre for Traffic and Transport
> > Building 115, 2800 Lyngby Denmark
> > Work: +45 45251536
> > Privat: +45 36308595
> > Fax: +45 45936412
> > E-mail: [log in to unmask]
> > Skype account: jeppe.husted.rich
> > ----------------------------------
> >
> > -----Original Message-----
> > From: Phillip Good [mailto:[log in to unmask]]
> > Sent: 17. februar 2004 15:36
> > To: [log in to unmask]
> > Subject: Re: distribution of 1/X , if X belongs to a normal distribution
> >
> > Yes, but what if none of the moments exist as in the present case?
> >
> > Phillip Good
> >
> > ----- Original Message -----
> > From: Osher Doctorow PhD <[log in to unmask]>
> > To: <[log in to unmask]>
> > Sent: Monday, February 16, 2004 11:52 PM
> > Subject: Re: distribution of 1/X , if X belongs to a normal distribution
> >
> >
> > > >From Osher Doctorow [log in to unmask]
> > >
> > > If Y is a function of random variable X, then under rather general
> > > conditions if fX(x) is the probability density function (pdf) of X
> > > for X continuous and if fY(y) is the pdf of Y for Y continuous, then:
> > >
> > > 1) fY(y) = fX(x(y))x'(y)
> > >
> > > where x'(y) is dx/dy evaluated at x(y) (the inverse of Y as a function
> > > of X all in terms of y rather than x).
> > >
> > > Rather than solve your particular problem, I'll give an example from
> > > Paul G. Hoel, S. C. Port, and C. J. Stone, Introduction to Probability
> > > Theory, Houghton-Mifflin: Boston 1971, pp. 120-121. Let X be exponent-
> > > ially distributed with parameter L, so that fX(x) = Lexp(-Lx) for
> > > L > 0 and 0 elsewhere. Find the pdf of Y = X^(1/b) where ^ indicates
> > > exponent. Since Y = X^(1/b), write y = x^(1/b), so x = y^b, and dy/dx
> > > = by^(b-1). Also, fX(x(y)) is fX(x) with x replaced by whatever
> > > function of y it is, or here x = y^b, so fX(x(y)) = Lexp(-Ly^b). Both
> > > factors are now functions of y, so just multiply them together as in
> > > equation (1) to get the answer when y > 0 (the density is 0 other-
> > > wise). Remember to take /fX(x(y))/, though.
> > >
> > > Osher Doctorow
> > > > Dear All,
> > > > I am an engineer, so please forgive me if my post seems to be
> > naive.
> > > > I am currently considering a very common structural problem. In
> > > > order to predict the behaviour of the response, I simulate the
> > system
> > > > using a Monte Carlo method. Without going into too much unecessary
> > > > detail, I am considering a system where the X random variable
> > follows
> > > > the Normal distribution ( X~N(\mu_X, \sigma_X) ), and another
> > random
> > > > variable Y is related to X with the following formula
> > > > Y= c/X,
> > > > where c is a constant.
> > > > As I said earlier, I am simulating this system and the
> > distribution
> > > > I am receiving for Y is not Normal (of course for low ratios of
> > > > (\sigma_X / \mu_X) it can be approximated by the normal
> > distribution ),
> > > > and is skewed. At this point, I am trying to avoid reinventing the
> > wheel
> > > > by trying to determine whether Y follows a formal statistical
> > > > distribution that I am not aware of. (I am aware of the inverse
> > normal
> > > > distribution, but I was led to believe that it is not suitable in
> > this
> > >
> > > > situation).
> > > > So I would appreciate it if you could provide me with
> > information
> > > > regarding the distribution that Y is following, if it follows one.
> > > > Also, if you could point me towards any publications that you feel
> > are
> > > > relevant I would appreciate it.
> > > > Thank you in advance.
> > > >
> > > > Kind regards
> > > > Nikos Papadakis
> > > >
> > > > --
> > > > Dr Nikolaos Papadakis
> > > > Research Fellow
> > > > IARC 4th Floor,
> > > > University of Warwick,
> > > > Gibbet Hill Road
> > > > Coventry CV4 7AL
> > > > tel: +44-2476 523684
> > > > fax: +44-2476 523387
> > >
> >
> >
>
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