How to solve such equations involving probability

Question

I am trying to understand how the equation (2) is obtained from the equation (1) that involves probability as shown in the following case:

$P_d = \text{Pr}\Bigl(h_d<h_e\Bigr)$ ----(1)

where $\text{Pr}$ denotes the probability, $h_d$ and $h_e$ are independent exponential random variables with variances $\sigma^2_{d}$ and $\sigma^2_{e}$ respectively.

$P_d = \frac{\sigma^2_e}{\sigma^2_e+\sigma^2_d}$ ----(2)

My query is how the equation (2) is obtained from equation (1).

Any help in this regard will be highly appreciated.

Answer 1

$P_d=\int_0^{\infty} \int_0^{y} \lambda e^{-\lambda x} dx \, \mu e^{-\mu y}dy$ where $\lambda$ and $\mu$ are the parameters of the exponential distributions of $h_d$ and $h_e$ respectively. Write down $\lambda $ and $\mu$ in terms of the variance and compute this integral.