Assume I have three scalar random variables $a,b,c \in \mathbb{R}$ related to each other through a multiplication operation:
$$ab=c$$
Let us further assume that we have three real-valued bounds for these variables:
$a \in [a_{min},a_{max}] \tag{1}$ $b \in [b_{min},b_{max}] \tag{2}$ $c \in [c_{min},c_{max}] \tag{3}$
Now assume that we take the limits for $c$ (3) as fixed. Under some circumstances, it is possible to tighten the bounds on $a$ (1) and $b$ (2) due to (3). However, I am not sure what the formal rules for this process are. Can you shed some light on the process behind this?
To illustrate, consider three cases:
Case 1: Let $a \in [2,10] $, $b \in [-20, -2]$, and $c \in [-8,-2]$. In this case, it is possible to tighten the bounds, and we can define tightened bounds $[2,4]$ for $a$ and $[-4,-2]$ for $b$.
Case 2: Let $a \in [4,4] $, $b \in [-10, 10]$, and $c \in [-4,6]$. In this case, it is possible to tighten the bounds for $b$ ($a$ is already a constant), and we have tightened bounds $[4,4]$ for $a$ and $[-1,1.5]$ for $b$.
Case 3: Let $a \in [-25, 25] $, $b \in [-25, 25]$, and $c \in [-10,10]$. In this case, it is not possible to tighten the bounds for $a$ and $b$.