So I came across below symmetry in my probability course that I can't understand.
I understand how the lower bound changes when removing the absolute value operator, but how does the 2 disappear?
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In the left hand you have the integral of a function of $|v|$ (considering the quadratic term too), hence that function is symmetric on $v$'s domain. This comes simply from the fact that $f(|-v|) = f(|v|)$.
Integrating that function on positive values of $v$ yields the same result as integrating it on negative values of $v$, so, to compute the whole integral, it is sufficent to only perform the integral on positive $v$ and to double it (if you don't you will not account for negative $v$).