Given the following CDF with $c \in\ \mathbb{R}^6, D_c:\mathbb{R}\to [0,1]$ and
$$ D(x):=\left(c_1 x+c_2\right) 1_{[-1,1)}(x)+c_3 1_{[1,2)}(x)+\left(c_4 \arctan \left(x+c_5\right)+c_6\right) 1_{[2, \infty}).$$
I have found the following restrictions for $c$ because of the characteristic properties of CDFs
- $c_1 +c_2 \leq 1$
- $0 \lt c_1 \lt c_2 $
- $c_1+c_2 \lt c_3 \lt 1$
- $c_3 \lt c_4 \arctan(2+c_5) + c_6 \lt 1 $
Now I have to find the probability measure of $D_c$. My first guess was that that the first summand will yield a Uniform-, the second one a Dirac- and the third one a Cauchy-Distribution because their density function should look pretty much like that. But how do I construct $P_c$ with the right jumps that could be between each Intervall and make sure that $P_c(\mathbb{R})=1$ ?