Confusing in transforming variable

[tonald]

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

 

[mathman]

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).