Random points on a circle

[alThor]

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For part b:
Okay, after thinking about it, common sense tells me that the correlation is -1, since they have a decreasing linear relationship... right? But how can I formalize this into a coherent answer?

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Write that relationship down! Replace for YX in the covariance (or corr.) formula, and just do the algebra from there. "Substitute and solve" to get the answer you think is right.

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For part c:
okay that kind of makes sense.

Would I want to do something like
P(p is in XY)*XY_bar + P(p is in YX)*YX_bar?


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That looks like you are trying to compute an expected value of some kind, so it's not quite right. If you prefer to go directly to the cumulative F(), ask yourself, before drawing Y and p, what is the probability that p will land in an arc sized less than half the circle? (There is always an arc sized less than (or equal) half the circle, but what is the probability p will end up in it, before knowing the exact size of that smaller arc? You can figure that out with an integral.)

alThor