I referred one of the research paper and found the expression of CDF. I am not getting how that CDF expression is derived. The different steps involved in the derivation are given below-
$I = \text{Pr}\Bigl(\alpha = 0\Bigr)$ ----(1)
where $\text{Pr}$ is the probability. $\alpha $ is given by $\alpha = \text{max}\Bigl(1-\frac{\phi}{P_s|f|^2},0\Bigr)$ where $|f|^2$ is exponential random variable and $P_s$ is signal to noise ratio (SNR), $\phi$ is a constant.
Finally, the equation (1) is derived as $I = \text{Pr}\Bigl(|f|^2<\frac{\phi}{P_s}\Bigr) = 1-\exp\Bigl(-\frac{-\phi}{\lambda P_s}\Bigr)$ ----(2)
I am not getting how the "=" sign in equation (2) changed to "<".
Any help in this regard will be highly appreciated.
Notice that: $$\alpha = 0 \iff 1 - \frac{\phi}{P_s|f|^2} \le 0$$
From this it follows: $$\mathbb P(\alpha=0) = \mathbb P \left(1 - \frac{\phi}{P_s|f|^2} \le 0\right) = \mathbb P \left(|f|^2 \le \frac{\phi}{P_s}\right)$$ You said in your question that $|f^2|$ is exponential, lets say with rate $1/\lambda$, then: $$\mathbb P \left(|f|^2 \le \frac{\phi}{P_s}\right) = 1-\exp\left(-\frac{\phi}{\lambda P_s}\right)$$