Consider a rectangle (black one) in the following image. Lets take four random points uniformly on each border then connecting the points one after another (red lines) to get a foursquare inside the rectangle.
If we put a set of random points ($n$ points) uniformly inside the rectangle , I would like to know what is the mathematical expectation of the number of points that are inside the red area?
Since the position of red points are random, I really can't solve this problem.
The probability that each point falls in the red area, is the area of red_line divided by area of rectangle. Since the area it self is a random process, so we need to calculate the expectation of the area of the red line.
Thanks in advance.