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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