I am having the following expression:
$$P = \text{Pr}\biggl(h_0 h_1 < \frac{u_1}{a_1 \rho - a_2 \rho u_1}\biggr)\tag{1}$$
where $h_0,h_1$ are independent and identically distributed (i.i.d) random variables, $u_1 = 2.4, a_1 = 0.8, a_2 = 0.2$ and $\rho$ is some positive constant.
Let $\phi = \frac{u_1}{a_1\rho-a_2\rho u_1}$, then eq. $(1)$ is written as
$$P = \text{Pr}\biggl(h_1< \frac{\phi}{h_0}\biggr);\quad \frac{a_1}{a_2}> u_1\tag{2}$$
$$P= 1;\quad \frac{a_1}{a_2}\leq u_1 \tag{3}$$
My query is that I am not getting how eq. $(2)$ and eq. $(3)$ are obtained on the basis of eq. $(1)$. Any help in this regard will be highly appreciated.
(3) doesn’t hold in general. For example set all the positive constants to 1 except u1=2. Then phi=-2. It is easy to define h0 and h1 such that their product is never less than -2 and indeed never negative, Eg as distributed according to a probability distribution with only positive support (Eg uniform on (1,2)).
Hope this helps