My Math Forum Conditional probability distribution change of variables

 July 27th, 2013, 11:41 AM #1 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 Conditional probability distribution change of variables Hello, I have a simple question regarding changing variables in a conditional distribution. I have two independent variables $r \in \mathbb{R}, r>0 t \in \mathbb{I}, t>0$ 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 $P(r;t) \mathrm{d}r$ which is "probability that the rate is r (within an interval \mathrm{d}r) at time t" This has a normalization condition $\int_0^\infty P(r;t) \mathrm{d}r= 1$ which means that there is some rate at any given time I am actually interested in finding a different conditional probability function $P(R;t)\mathrm{d}R$ where R is the cumulative rate up to time t So if if have outcomes for $t= 1,2,3,4,\dots,t_{f}$ that are $r_{1},r_{2},r_{3},r_{4},\dots,r_{t_{f}}$ then the cumulative rate is $R_{t_{f}}=\mathrm{ \Pi}_{i=1}^{t_{f}}r_{i}$ , which is to say the product of $r_{i}$ for i up to $t_f$ Again, this would have to have a normalization condition $\int_0^\infty P(R;t) \mathrm{d}R= 1$ since for any give time there has be some cumulative rate. If you can help me find $P(R;t)\mathrm{d}R$ from $P(r;t)\mathrm{d}r$ I would greatly appreciate it. Thank you very much for you help. Ilya
 July 27th, 2013, 12:59 PM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 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.
 July 27th, 2013, 02:48 PM #3 Newbie   Joined: Jan 2012 Posts: 3 Thanks: 0 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 $N^2$ 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).

### change of variables probability forum piecewise

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Andrew.F Calculus 0 September 17th, 2012 05:08 PM jigsaw Advanced Statistics 3 August 18th, 2012 12:26 PM meph1st0pheles Advanced Statistics 1 March 23rd, 2010 05:47 PM edspecks2004 Advanced Statistics 0 August 18th, 2009 03:11 AM tmac3813 Calculus 5 December 17th, 2008 05:39 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top