Uniform distribution find E(Y|x)

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To find E(Y|x) for a uniform distribution over the circle defined by x^2 + y^2 ≤ 9, it is essential to understand that E(Y|x) represents the mean value of Y for a fixed value of X. The integration should be performed with respect to Y, while X remains constant, and Y ranges from -√(9 - x^2) to √(9 - x^2). A correct approach involves integrating Y over this range to determine the expected value. The final result for E(Y|x) is expected to be zero, which aligns with the symmetry of the distribution.
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This is the question:

If X and Y have a uniform distribution over the circle x^2 + y^2 \leq 9 find E(Y|x).

Can someone please explain to me, how to answer this question. You guys don't have to give me a solution, but a hint would be nice because I have no idea where to start. Thank you :smile:
 
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well what i was thinking was that the range is between 3 and 0 and -3 and 0.

Then you integrate x^2 + y^2 with the first range (3 and 0) and then with -3 and 0. Is this right?

Or is the first range y to 3, and then -3 to 0?

I have no idea, please help me
 
"E(y|x)" means the mean value of y for a single value of x. There will only be an integral with respect to y, not x. x is fixed. y ranges between -\sqrt{9- x^2} and \sqrt{9- x^2}. Your final answer for E(y|x) will be a function of x.
 
Thank you for replying. Yupp I think that i got the answer. I got zero at the end, but I'm pretty sure that's right
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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