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Expected value of area

  1. Mar 25, 2008 #1
    Given a unit square and a 'target' square of size LXL = p^2 < 1 in the unit square. The center of target square in equally likely to be anywhere in the unit square. What is the average size of the target square as a function of p^2.

    This is the problem and I have included a jpeg illustration of the problem. Any help would be greatly appreciated.

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  2. jcsd
  3. Mar 25, 2008 #2
    The size of the "target square" is p^2 by the statement of the problem. :smile:

    Do you mean the intersection of the target square T with the unit square [itex]U=[0,1]\times[0,1]\subset \mathbb{R}^2[/itex]? In this case I suggest you first try and find out what the area of [itex]A = T\cap U[/itex] is as a function of the center of the target square. Denote this center by (x,y).

    Then for [itex]p \leq x\leq 1-p \wedge p \leq y\leq 1-p[/itex] you have A=p^2. Figure out the area A if the center of the target space is too close to the boundary of the unit square for the former to completely fit into the latter.
    Last edited: Mar 25, 2008
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