This is one of the classic problems in geometric probability. It (and many variations) are "covered" (pardon the pun) in books like Hall (1988) "Introduction to the theory of coverage processes" and Solomon (1978) "Geometric probability". However, I think you should first check an old paper by Cyril Domb (1947) "The problem of random intervals on a line" Proc. Cambr. Phil. Soc., vol 43, 329-341. If I remember correctly, this paper deals with the problem you describe.

Alireza, yes there are many cases. Just take a Pareto´s distribution for descense ordering of K variable. Its Lorenz curve has shape L(p)=p^c (p=cummulative population, 0

Thank you guys. Dear Michaela and Emilio! I can't get the details of your comments. Please write the references of your explanations.

Dear Michael. I got the Solomon book. But I couldn't find other references. I will be happy you tell me how can I get the Cyril Domb's paper.

Thanks

Statistics of the Lp-norm may help in a general context.

The following book is not directly related to your question, but is a fantastic source on related topics: B. Schweizer and A. Sklar. Probabilistic Metric Spaces. Elsevier Sciences, 1983. Reprinted by Dover in 2005.

I reformulate in this way: let (X,d) a metric space, and P a probability measure on the measure space (X,B(X)), where B(X) is the Borel sigma-algebra. Defined then the product probability space (X*X,B(X)*B(X),P*P), the quantity you are requesting is given by P*P{(p,q) in X*X|d(p,q)

Dear A. A. J. Marley My master thesis was in probabilistic metric and normed spaces. My question becomes from this field. we know that F(p,q)(t) is the probability of the distance between p and q. So my question is: what is this probability distribution function in a specific space?

Dear Andrea Andrisani Thanks for reformulation. But the question isn't solved.

Dear Pietro Piu, Thank you. I am trying to understand the concept of your comment. probably the Weibull distribution is useful.

Alireza, If I got your question correctly, you could find the pdf of the order statistics X(i), and the joint pdf of X(i) & X(j), then you can find the pdf of the distance X(i)-X(j); i>j.

Dear Mahmoud, suppose we have the pdf of X(i) and X(j), the problem is: how do we can find the pdf of d(x(i),x(j))?

Alireza, I understood better your question from your post to Mahmoud. The question can not be answered from the pdf -because inverse function has been only defined with integrals using a tautologic theoretical formula based in PDF that gives probability as a graphic shaded area and not as a value-. A solution may be found from the CDF that gives the minimum X(i) value for any cumulative population fraction P, when descending order of X is used. That is the reason I tried to explain it using Pareto's distribution where cumulative population P is clearly defined as a function of X value. I have tried several times to explain this in RG questions and answers using a non-traditional, non linear method based in Laplace Criterion with a Lorenz Curve model of shape L(p)=p^(W(p)) and the basic SEM needed is W(p). I have written papers about this in spanish language, but I am now in the process of writting a paper in english language about it, this will take some time more and people like Demetris is kindly helping me in this task. Thanks, emilio

Alireza, An answer for your question is as follows:

Let X 1X2X n, denote the order statistics of a random sample, X 1 , X 2 , ......, X n from a discrete population with c.d.f F X x and p.d.f f X x. Then the joint c.d.f of Xi:n and X j:n , 1 i j n, is

F_(i,j:n) (x_i,x_j )=∑_(s=j)^n▒∑_(r=i)^s▒n!/r!(s-r)!(n-s)! [F_X (x_j)]^r [F_X (x_j )-F_X (x_i)]^(s-r) [1-F_X (x_j)]^(n-s),

for i j.

The joint probability mass function of Xi:n and X j:n (1i j n ), is:

f_(i,j:n) (x_i,x_j )=P(X_(i:n)=x_i,X_(i:n) )=

F_(i,j:n) (x_i,x_j )-F_(i,j:n) (x_i-1,x_j )-F_(i,j:n) (x_i,x_j-1)+F_(i,j:n) (x_i-1,x_j-1)

Reference:

Calik S. (2005),"On the Variances of Distribution of the Sample Range of OrderStatistics from a Discrete Uniform Distribution", Journal of Mathematics and Statistics 1(3): 180-183.

Dear Mahmoud, Thank you for your attention. I couldn't find any probability of Distance in that paper and your comment. In fact I didn't look for joint probability mass function. I just want to know the probability that the distance between x(i) and x(j) is less than t.

Dear Emilio, Thanks for your answers. Do you mean I should wait for your English paper? Is there any way to simply I find my answer?

Dear Alireza, Yes, my last comment was just an answer to your latest question on finding the joint density of two order statistics. But next step should be to use this joint density in finding the c.d.f of Y(i) = x(i) - x(j) which is P[(x(i) - x(j))< t] which I hope answers your question.

Guys! Let me explain my problem. This problem comes from the applications of probabilistic metric spaces. The notion of probabilistic metric spaces was introduced by Menger. The idea of Menger was to use distribution functions instead of non-negative real numbers as values of the metric. It corresponds to the situations when we do not know exactly the distance between two points, we know only probabilities of possible values of this distance. So when we have a probability space with variables as points, how do we find the the probability that the distance between two points is less than real number t ?

Dear Alireza. It is easier for me to make the estimations than to explain them. If you agree, put here a dataset in excel format of positive univariate values with N=100 or N=20, as you prefer. I will work it and send you an excel table with results, so you may interpret and comment it. If you have critics and questions, make them, because they help me to understand my limits and those of the model and to work them. It may also help you to compare it with other results and methods applied to same dataset, something which is desirable in any analysis. OK, thanks and good luck, emilio

Dear Emilio, actually I am a mathematician and I am working on a theoretical problem. I have no data set right now. I should finish my theoretical work and then verify it by natural data. I expected some body help me in this theoretical problem. maybe I couldn't explain my Idea!

Distance between random points, "random" defined in many ways, can be found in my book on geometrical probability "An Introduction to Geometrical Probability: Distributional Aspects with Applications", Gordon and Breach, Newark (1999)

Alireza, you explained your points very well. I attach here a diagram of one page that summarizes the mathematical/numerical method I use. It is part of a working paper not finished yet. Perhaps it may help, perhaps not. Thanks, emilio

Dear professor Arak Mathai Mathai , I couldn't find your book. I can't buy it from internet.

Thank you for your suggestion.

Thank you Emillio.

I am trying to read it.

I actually have a paper on fuzzy sets that looks quite close to your problem. What if I would like to find the barycenter of a sets of fuzzy points. The distances would be fuzzy themselves and also the barycenter. I don't know if it can lead you to something but check:

http://www.igi-global.com/article/the-weighted-fuzzy-barycenter/102944

If you cannot download it, please, let me know to send it directly to you.

I couldn't download it. please send me a copy of this article. Thank you Julio

Dear Alireza,

The following link will give you an example of probability distribution of the distance between two random points in a box.

        http://www.math.kth.se/~johanph/habc.pdf

In addition, C. Alsina, M.J. Frank and B. Schweizer’s book Associative Functions (Triangular Norms and Copulas) will help you the relations between related concepts in this area. At least, the example given on the foregoing link may be a starting point for you.

Good Luck..