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July 27th, 2013, 11:41 AM   #1
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Conditional probability distribution change of variables


I have a simple question regarding changing variables in a conditional distribution.

I have two independent variables

where r is "rate" (can be any positive real number although most likely to be around 1) and t is "time" (positive integers ie 1,2,3,4...).

I have a conditional probability function (really a probability density function) of the form

which is "probability that the rate is r (within an interval \mathrm{d}r) at time t"

This has a normalization condition

which means that there is some rate at any given time

I am actually interested in finding a different conditional probability function

where R is the cumulative rate up to time t

So if if have outcomes for that are then the cumulative rate is , which is to say the product of for i up to

Again, this would have to have a normalization condition

since for any give time there has be some cumulative rate.

If you can help me find from I would greatly appreciate it.

Thank you very much for you help.

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July 27th, 2013, 12:59 PM   #2
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Re: Conditional probability distribution change of variables

Hi Ilya,

I think that might be very difficult to do for a general pdf P(r). I tried it with P(r) constant between 1/2 and 3/2 and worked out the pdf for R, the product of the first two r1 and r2. It was:

P(R) = 2R - 1/2 (for R in [1/4, 3/4])
P(R) = 3/2 - 2R/3 (for R in [3/4, 9/4]

You might be able to generalise this for the product of the first n rates for this or other simple pdf's. It would be good, therefore, if you knew what your pdf P(r) is.
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July 27th, 2013, 02:48 PM   #3
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Re: Conditional probability distribution change of variables

The problem is that the pdfs are obtained numerically from empirical data and are not "nice".

One thing I can do if I can't find an analytic approach is to do it numerically. I will take N rates which are described by P(r,1) and cross-multiply with N rates described by P(r,2). I will then re-bin the resulting rates back to N rates and obtain a pdf (this will be my P(R,2)). I will then cross-multiply that by N rates from P(r,3) and then re-bin to find P(R,3) and so on. The problem is that I fear N will have to be quite big for this to work, I need to go for somewhat large times, and I have thousands of different processes (each with their one P's) for which I will need to do this. So all in all it would be quite computationally expensive (for N= 1000, max t=20, and 1000 different processes I would need to do 20 billion multiplications and 20 thousand rebinnings).
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