I have an expression that gives me the bounds for a certain variable X in terms of another variable Y. How can I write these bounds as a function of bounds of Y?
Specifically, if $$X \in \lbrack max(0,Y-1), min(1,Y)\rbrack$$ and $$Y \in \lbrack a, b\rbrack$$ How do I find f(.) and g(.) in the expression below? $$X \in \lbrack f(a,b), g(a,b)\rbrack$$
Note that:
- X is a probability, so $X\in\lbrack0,1\rbrack$
- Y is sum of 2 probabilities, so $Y\in\lbrack0,2\rbrack$. This does not mean that a=0 and b=2 but that $\lbrack a,b\rbrack \subseteq \lbrack0,2\rbrack$
You have, rewriting the first two bounds you have: $$ \begin{align} X \geq Y-1, \ X\geq 0 \\ X \leq 1,\ X \leq Y \\ a\leq Y \leq b \end{align} $$
so that $$ a-1 \leq Y - 1 \leq X \leq b $$ and overall $$ \max(a-1, 0) \leq X \leq \min(1,b). $$