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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