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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}}2$, $\beta>0$, $m>0$.

I tried to solve this by first finding pdf of $d_r$ given $ \frac{1}{d_r^{\alpha}}

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    I don't know what the context of the question is, but it looks like the bound was calculated with Chebyshev's inequality: https://en.m.wikipedia.org/wiki/Chebyshev's_inequality. – 2017-01-09
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    @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? – 2017-01-09
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    @Michael Any idea how to solve this? – 2017-01-10
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    Which PDF (or CDF) of $d_r$ did you find? – 2017-01-10

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