Confusing in transforming variable

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    Confusing Variable
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SUMMARY

The discussion centers on the transformation of variables involving two independent uniform random variables U1 and U2, both distributed as Uniform(0, A). The derived variable x = (U1 + U2) / 2 results in a probability density function (PDF) that is not uniform, but rather forms an isosceles triangle with a peak at x = A/2. The confusion arises from the integration of the PDF over the range from 0 to A, which yields a total probability of 2 instead of the expected 1, indicating a misunderstanding of the density function's shape and properties.

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  • Basic calculus skills for integration
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tonald
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Considering U1 and U2 which both follow Uniform (0 , A), then after
the transformation of variable, i can find x=(U1+U2)/2 has f(x)=2/A(both
my classmates and me get this result). However i am confused that when i
integrate x by it range from 0 to A (because U1 and U2 both ranged from
0 to A), the prob equals 2 instead of 1. What's wrong with my mind?
steps : http://desmond.imageshack.us/Himg848/scaled.php?server=848&filename=steps.png&res=landing
 
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I can't follow your derivation, but the answer is completely wrong.
Let U=(U1+U2)/2. The density function for U is not uniform.
It looks like two sides of an isosocles triangle with the base from 0 to A on the x-axis and the peak of the triangle is at x = A/2. The two sides have equations: y=(4/A2)x and y = (4/A2)(A - x).
 

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