I have a decision (detection) problem trying to decide between symbols ${0,2}$. I have the two probability density functions: $ f(z|s=0) = \begin{cases} 0.25z + 0.5, & -2\le\ z <0 \\ -0.25z + 0.5, & 0\le\ z \le\ 2 \end{cases} $
and $ f(z|s=2) = \begin{cases} 0.25z, & 0\le\ z <2 \\ -0.25z + 1, & 2\le\ z \le\ 4 \end{cases} $
How can i mathematically prove that the optimal threshold value $T$ for that decision problem is equal to $1$?