density of integral(RV(t)~f(t), 0..T, dt)
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? 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)) I'd greatly appreciate answers to any of these questions or any references that might deal with this problem. Thanks, Thomas Burg Dept. of Physics, Swiss Federal Institute of Technology [EMAIL PROTECTED] === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
density of integral(RV(t)~f(t), 0..T, dt)
Can't be done without knowledge of the joint distributions of Y(t1), Y(t2),..., Y(t). Jon Cryer --- Text of forwarded message --- X-Authentication-Warning: jse.stat.ncsu.edu: majordom set sender to [EMAIL PROTECTED] using -f To: [EMAIL PROTECTED] Date: Wed, 19 Apr 2000 16:46:32 +0200 From: Thomas Peter Burg [EMAIL PROTECTED] Organization: University of Illinois at Urbana-Champaign Reply-To: [EMAIL PROTECTED] Subject: density of integral(RV(t)~f(t), 0..T, dt) Sender: [EMAIL PROTECTED] Precedence: bulk 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? 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)) I'd greatly appreciate answers to any of these questions or any references that might deal with this problem. Thanks, Thomas Burg Dept. of Physics, Swiss Federal Institute of Technology [EMAIL PROTECTED] === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ === _ - | \ Jon Cryer[EMAIL PROTECTED] ( ) Department of Statistics http://www.stat.uiowa.edu\ \_ University and Actuarial Science office 319-335-0819 \ * \ of Iowa The University of Iowa dept. 319-335-0706\ / Hawkeyes Iowa City, IA 52242FAX319-335-3017 | ) - V === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
Re: density of integral(RV(t)~f(t), 0..T, dt)
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 === This list is open to everyone. Occasionally, less thoughtful people send inappropriate messages. Please DO NOT COMPLAIN TO THE POSTMASTER about these messages because the postmaster has no way of controlling them, and excessive complaints will result in termination of the list. For information about this list, including information about the problem of inappropriate messages and information about how to unsubscribe, please see the web page at http://jse.stat.ncsu.edu/ ===
