Vrftsjtryk

I am currently studying probabilities while doing some game theory, and I'm having trouble solving a particular question that I came up with.

Suppose you have two random variables, $sigma_1$ and $sigma_2$, where $sigma_1$ and $sigma_2$ are independently distributed. Suppose further that $sigma_1$ is uniformly distributed in an interval $[-phi, phi]$ and $sigma_2$ is uniformly distributed in an interval $[0,bar{sigma}_2]$ (not too important here).

Now, let there be some constants $a,b,c$.

How would we calculate this probability?

$p(2b-sigma_1 +2c geq 2sigma_2)$ and $p(b-sigma_1 geq a)$

Here's what I have so far. Let $mathbb{1}$ denote the indicator function and let $F(sigma_1)$ and $G(sigma_2)$ represent the distribution functions:

$int_{sigma_1} int_{sigma_2} mathbb{1} [sigma_1 leq min{2b+2c-2sigma_2, b - a}] dG(sigma_2) dF(sigma_1)$

Assuming that $bar{sigma}_2$ is large enough ($bar{sigma}_2 > b-a$), can I write this as:

$int_{sigma_1} left[int_{b-a}^{bar{sigma}_2} mathbb{1}[sigma_1 leq b - a] dG(sigma_2) + int_{0}^{b-a} mathbb{1}[ sigma_1 leq 2b+2c-2sigma_2] dG(sigma_2)right] dF(sigma_1)$

Is this correct? Or can anyone help clarify how to simplify this?

Thank you