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Assume we have a box shaped area $(x,y)\in ([0,1],[0,1])$. There are $n$ kings $K_1, K_2,\cdots, K_n$ placed in this box randomly and independently according to uniform distribution. Enemies $E_1, E_2, \cdots$ are located in this area according to 2 dimensional Poisson point process with average $\mu$. Also, Queen is located at the point $(x,y)=(1,0.5)$. Show that for large enough $m$ $$E\left[\frac{1}{d_r^\alpha}\bigg| \frac{1}{d_r^{\alpha}}<m^{\alpha}\right] \leq \beta{ m^{2-\alpha}}$$ where $E[\cdot]$ is the expected value all possible locations of Kings and Enemies, $d_r$ is the distance between the closest enemy to the king $K_r$, $\alpha>2$, $\beta>0$, $m>0$.

I tried to solve this by first finding pdf of $d_r$ given $ \frac{1}{d_r^{\alpha}}<m^{\alpha}$, and then taking expectation. But, that is a huge mess. Any shortcuts? Any idea?

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  • $\begingroup$ I don't know what the context of the question is, but it looks like the bound was calculated with Chebyshev's inequality: en.m.wikipedia.org/wiki/Chebyshev's_inequality. $\endgroup$ – Cehhiro Jan 9 '17 at 8:39
  • $\begingroup$ @O.VonSeckendorff Thanks for your comment. I have worked with Chebyshev a lot. But that is useful when you want to upper bound a probability. I am not sure how that can be applied to the expectation above. Can you clarify how generally it can be used to upper bound expectation? $\endgroup$ – Susan_Math123 Jan 9 '17 at 17:16
  • $\begingroup$ @Michael Any idea how to solve this? $\endgroup$ – Susan_Math123 Jan 10 '17 at 0:30
  • $\begingroup$ Which PDF (or CDF) of $d_r$ did you find? $\endgroup$ – Did Jan 10 '17 at 16:50

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