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.