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#1




SOA Sample Question #171
Hello,
I am struggling with this question. I don't quite understand the solution provided for this question. It uses some properties of the uniform distribution. I tried to determine E[X] and E[X^2] by breaking the bounds and integrating the joint density over the appropriate bounds. Thus, finding out the variance of X. However, I don't think I have split the bounds accurately as I have ended up with some weird answer. I believe what's really confusing is the modulus surrounding the x and y variables. Any help is appreciated! The correct answer is A) 1/6. Thank you for all your help! 
#2




The distribution is symmetric about 0. The mean of X is zero. You can calculate the variance as the second moment. Because of the symmetry about zero, the second moment can be calculate as the conditional second moment where both X and Y are greater than zero with f(x,y  x >0, y > 0) = 2.

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absolute values, bounds, joint distribution, limits, modulus 
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