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Expected value of a semicircle

  • Thread starter de1337ed
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  • #1
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Homework Statement


Point is chosen at random (uniform PDF) within semi-circle: {(x,y)|x2+y2≤r2, y≥0}

Basically, I'm supposed to find E[X] for this problem

2. The attempt at a solution

I know how to do it, in a very long-winded fashion
(find fY(y), and E[X|Y=y] etc).
But my teacher says that there is an easier way that doesn't involve a lot of calculations, anyone see it?
Thanks

EDIT;;;;

I think i might have gotten it, because the semicircle is an even function, the expected value of the first moment is 0. Can someone confirm this also?
Also, does this apply to a regular circle as well?
 
Last edited:

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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the semicircle is an even function
That is not a precise statement. What function are you referring to? If you are referring to the equation for the boundary of the semi-circle, that doesn't justify any conclusions about the moment of X.
the expected value of the first moment is 0
You could argue by symmetry if the distribution function that the mean value of X in the semicircle is 0. ( Don't say "expected value of the first moment" because the first moment of X is the expected value of X.)
 

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