The notation is confusing. I presume when you say P(r,t) you don't mean a joint probability distribution on r and t but rather a probability density on r that's parameterized by the observation time t? Usually that would be notated as Pt(r) or P(r|t) or something like that to avoid the implication that r and t are the same types of variable. If one is a random variable and the other is a parameter, then P(r,t) is a very confusing way to write it. What would it even mean for the observation time to be a random variable? What random process is producing it? It really makes more sense if t is a parameter. But I can't be sure of this interpretation of you question.

Changing the notation to what I think you meant, P(r|t)dr is the probability to find the distance within (infinitesimal) dr of the value r, given that the observation time is t.

Now, by P(k,r) do you mean that to be a joint probability distribution in which both k and r are random variables, or a conditioned probability distribution in which k is a random variable and r is a parameter? The latter seems more likely, since there's no observation time specified and the thing would therefore be dominated by large times, where presumably there's no bound on r, which would mean the joint probability distribution probably isn't normalizable.

So I'm presuming that when you say P(k,r) you mean a conditioned probability density, which I would write as P(k|r). Now, you say this function is known, and I note that there's no observation time t in it at all. Are we, then, allowed to assume that P(k|r) is independent of time? Or is it averaged over time--which would imply that it's the large-time asymptotic distribution, since again, the average is going to be dominated by large times since there's apparently no upper bound on t. In the latter case I don't think you have enough information to calculate it.

But in the former case, if you know that P(k) depends only on r and not directly on t (given a value of r), and P(r) depends only on t, then you just integrate over all possible r values:

P(k|t) = Integral(P(k|r)*P(r|t)dr)

But this is only one possible answer, depending on what the real situation is; I had to assume quite a few things about your question to produce that.

I think it is important to know whether the random variables are independent or not. In case they are independent, the joint PDF f(r,t) = f(r). f(t) also f(k,r)= f(k). f(r) accordingly f(k,t) = f(r,t) . f(k,r) / (f(r))^2, however you still need to know f(r). I hope this is helpful a little bit.

Dear Akram,

Thank you for your answer. I think that the two variables are not independent. For P(r,t), the more the time is large, the more the distance r that separates the two particles increases. The same goes for P(k,r) because the more the distance is large the less the pair will rotate, We can also anticipate that the variables k (rotation rate) and t (time) are not independent because we expect that the pair will rotate slower at later times due to the increase in the separation r. By the way, I don't know whether the PDFs given above are joined PDFs or they are marginal distributions?

Cheers,

The notation is confusing. I presume when you say P(r,t) you don't mean a joint probability distribution on r and t but rather a probability density on r that's parameterized by the observation time t? Usually that would be notated as Pt(r) or P(r|t) or something like that to avoid the implication that r and t are the same types of variable. If one is a random variable and the other is a parameter, then P(r,t) is a very confusing way to write it. What would it even mean for the observation time to be a random variable? What random process is producing it? It really makes more sense if t is a parameter. But I can't be sure of this interpretation of you question.

Changing the notation to what I think you meant, P(r|t)dr is the probability to find the distance within (infinitesimal) dr of the value r, given that the observation time is t.

Now, by P(k,r) do you mean that to be a joint probability distribution in which both k and r are random variables, or a conditioned probability distribution in which k is a random variable and r is a parameter? The latter seems more likely, since there's no observation time specified and the thing would therefore be dominated by large times, where presumably there's no bound on r, which would mean the joint probability distribution probably isn't normalizable.

So I'm presuming that when you say P(k,r) you mean a conditioned probability density, which I would write as P(k|r). Now, you say this function is known, and I note that there's no observation time t in it at all. Are we, then, allowed to assume that P(k|r) is independent of time? Or is it averaged over time--which would imply that it's the large-time asymptotic distribution, since again, the average is going to be dominated by large times since there's apparently no upper bound on t. In the latter case I don't think you have enough information to calculate it.

But in the former case, if you know that P(k) depends only on r and not directly on t (given a value of r), and P(r) depends only on t, then you just integrate over all possible r values:

P(k|t) = Integral(P(k|r)*P(r|t)dr)

But this is only one possible answer, depending on what the real situation is; I had to assume quite a few things about your question to produce that.

Let p(r,t), q(φ,r) and s(φ,t) be the relevant probability distribution functions. The sought-after probability distribution function s(φ,t)  is equal to the integral with respect to r of the product q(φ,r) p(r,t). Note that since at a given t the two particles are somewhere, the integral with respect to r of p(r,t) is equal to unity, independent of t. Similarly, the integral with respect to φ of both q(φ,r) and s(φ,t) is independent of respectively r and t and equal to 1. One can readily verify that the expression for s(φ,t) that I have presented above satisfies this requirement.

Dear Bryan&Behnam,

I would like to thank you for your comments, they are very helpful.

Indeed, as Bryan mentioned in his comment, we may write Pt(r) for the probability of finding the pair at a given separation r at a later time t because t is just a parameter. The same thing for Pr(k) because we look to the probability to find the pair at the rotation rate k when a given separation r is reached.

Regarding the dependence on time of Pr(k), the PDFs are computed without taking time into consideration. I mean, whenever a separation r in found, we find out the PDF of the rotation rate k. I hope we are still able to apply the formula you mentioned for Pt(k).

Thank you,

You are welcome Abdallah.

I agree with Dr. Bryan, I think his comment is complete. 

By adding, multiplying Probability density function, or by changing the definition fiel of function et coéputing the related marginal Law.

@Yacin Could you please be more precise? An example would be highly appreciated, thanks!