It is not clear, to me anyway, whether you want: 1) The probability that two intervals chosen "at random" will have lengths within the chosen constraints and will intersect; or
2) The probability that two intervals chosen "at random" from among those that satisfy the constraints will intersect. Ian. "Cristian-Augustin Saita" <[EMAIL PROTECTED]> wrote in message as4nh9$kk8$[EMAIL PROTECTED]">news:as4nh9$kk8$[EMAIL PROTECTED]... > Hello, > > I need to compute the intersection probability of two intervals randomly > chosen in the domain [0,1] such to respect some constraints related to > their sizes. Precisely, the size of the first interval has to be between > a and b, and the size of the second interval between c and d, > where 0 <= a <= b <= 1 and 0 <= c <= d <= 1. > > I've managed to compute this probability for the simple case where no > size restriction is imposed to any of the two intervals. It gives 2/3 = > 0.666... which verifies experimentally. > > I've also managed to compute the intersection probability when the size > of the first interval is restricted between a and b, while the size of > the second interval has no restriction. Obviously the corresponding > formula depends on a and b: > > P = (2*a^3 + 2*b^3 - 4*a^2 - 4*b^2 - 4*a*b + 3*a + 3*b + 2) / (2 - > a - b) / 3 > > I've experimentally verified this formula and it works well. > > Nevertheless, my computing method was mostly intuitive based on the > discretization of the domain [0,1] in N regions and on the > representation of the intervals as ordered pairs of numbers between 1 > and N. Considering such interval pairs, I counted and expressed as > functions of N: (1) the number of combinations of interval pairs > overlapping each other; and (2) the total number of possible interval > pair combinations. Considering N -> infinit, the fraction between the > two terms (1) / (2) gave me the intersection probability . > > Unfortunately, my counting method becomes too complex when also > restricting the size of the second interval to [c,d]. > > I know that such problems could be better expressed as integrals in a > suitable multi-dimesional data space (probably 4-dimensional in this > case), but this is not trivial to me. > > Could someone help me to express the interval intersection probability > in a mathematical form, such to compute the general formula depending on > a, b, c and d? > > Thank you very much, > > Cristian Saita > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
