Originally Posted by
nicola I'm thinking of the following problem. I'd appreciate if you share your thought about my solution,
or correct it. There is no answer for the problem.
Problem: There is a square with size of a^2 on 2 dimensional spaces. We imagine lines that touch the square. If we describe the line with x (length) and t (angle from horizon), then each line for the parameters x and t is associated with the probability f(x,t).
Question is, what is the average number of lines that touch the square? Range of x : from b to c. Range of t: from 0 to 2*pi.
I'm thinking following solution:
For the fixed x and t, if the line should touch the square, one end of the line should be in the polygon with the size {sqrt(2)*a*x + a^2}.
Thus the answer (for every x and t) is:
int int_R {sqrt(2)*a*x +a^2}* f(x,t) dA ( int: integral )
= int_b^c int_0^(2*pi) {(sqrt(2)*a*x +a^2}* f(x,t)* x dt dx.