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/ . =================================================================
