In article <[EMAIL PROTECTED]>,
Thomas Peter Burg <[EMAIL PROTECTED]> wrote:
>Does anyone know if there's an answer to the following problem:
>I'm given a function of time Y(t), with the property that all values of
>Y are
>random variables which are drawn from a time dependent distribution with
>known time dependent density f(t). I.e. the probability that Y(t)>x is
>Integral(f(t),-inf..x,dt):
>d/dx P( Y(t) > x ) = f(t)
>With these facts given, is there anything that can be said about the
>distribution of
>Integral(Y(tau), 0..t, dtau) ??
>or its density function?
With the information given, all that can be stated is that
the expected value of the integral is the integral of the
expected value.
The Y(t) had better be independent, or if the integral
makes sense, it is going to be a constant almost surely.
So to do anything with the distribution, it will be
necessary to know the nature of the dependence.
>Is there a nice expression for that in terms of the known density f(t)
>in
>general?
>or maybe with specific assumptions about f? (E.g. Gaussian with mean(t)
>and
>var(t))
One also would need the covariance cov(t,u). If there
is a reasonable amount of measurability, the variance
of the integral is the double integral of the covariance
function.
If the joint distributions are normal, the integral will
also be normal. But one still needs the covariance
function, not just the variances.
However, for general distributions, more information is
needed to do anything about the distribution. Simple
answers are not always forthcoming.
--
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, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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