Let $\gamma_1,\gamma_4$ and $\gamma_f$ be three positive independent random variables with PDFs $f_{\gamma_1}(\gamma), f_{\gamma_4}(\gamma)$ and $f_{\gamma_f}(\gamma)$, respectively.
The CDFs are denoted by PDFs $F_{\gamma_1}(\gamma), F_{\gamma_4}(\gamma)$ and $F_{\gamma_f}(\gamma)$, respectively.
Let $\beta$ be positive number.
I was have the following problem : $$ P\left\{(\gamma_4+\gamma_f\leq \beta) \cap (\gamma_1<\gamma_f) \cap(\gamma_4\leq \gamma_1)\right\}. $$ Then I simplify to :
$$ P\left\{\gamma_4\leq \gamma_1<\gamma_f<\beta-\gamma_4\right\}. $$ I would like using the PDFs and CDFs to compute the following probability: $$ P\left\{\gamma_4\leq \gamma_1<\gamma_f<\beta-\gamma_4\right\}. $$
Is it $$ P\left\{\gamma_4\leq \gamma_1<\gamma_f<\beta-\gamma_4\right\}= \left\{\int_{y_4=0}^{\beta/2}f_{\gamma_4}(y_4) \left(\int_{y_f=0}^{y_4-\beta}f_{\gamma_f}(y_f)\left[\int_{y_f=0}^{y_f}f_{\gamma_1}(y_1)dy_1\right]dy_f\right)dy_4\right \}. $$ Thanks.