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I'm kinda new to stats and trying to find info on PDFs that depend on multiple parameters, but I keep finding info only on multi-variate distributions.

The point is that I have only one random variable $R$, but it depends on two or more parameters, say $(x_1,x_2,x_3)$. My actual goal is: I know the dependency of R on the separate params, and I need to create a joint PDF for it.

An example can be: say you have the distribution of heights based on age, and separately based on weight, how do you combine those to one PDF such that given (age, weight) will give the correct height distribution? Any help will be very appreciated!

Edit: it seems that there's not enough data to “build” a joint PDF. Will the answer be any different if I can assume everything is distributed normally? Meaning, can the “joint” mean and variance be determined given age + weight?

MarianD
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Steve Jolt
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  • it is not possible, there are too many unknowns. – Xi'an Dec 21 '20 at 08:46
  • The answer to the edited question also is in the negative, as beautifully illustrated and explained at https://stats.stackexchange.com/questions/30159. – whuber Dec 21 '20 at 12:21

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If you only have the age and weight density, you cannot do much else than multiply them, and then you end up with the same age distribution for every weight for example, which is probably not what you are looking for.

The problem is that how weight differs for multiple age groups is not contained in either distribution. So you don't have any information on that to start with.

Lets say age ~ uniform(0, 100) and weigth ~ N(60, 20). Then neither of those can tell you that young people are lighter. So you'd need additional information.

Gijs
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  • I think you maybe missed the point? I'm looking at height, I don't know how age or weight distribute! But I know how height is distributed according to both. What I'd like to do it have a PDF for height given (age, weight) pair as input. I hope this makes sense – Steve Jolt Dec 21 '20 at 09:12
  • Ah yes, that's a bit different, but the problem is the same actually. – Gijs Dec 21 '20 at 09:21