Regarding transformations of random variables, there is a problem in my textbook that is interested in a transformation $U_2=Y_1Y_2$ where $0<y_1<1$ and $0<y_2<1$.
The book then says that if we consider the function $u_2=y_1y_2$ then this function alone is not a one to one function of the variables $(y_1, y_2)$. So instead consider $$u_1=y_1, \quad \text{and} \quad u_2=y_1y_2.$$ For this choice of $u_1$ and $0<y_1<1$, $0<y_2<1$ the transformation from $(y_1, y_2)$ to $(u_1, u_2)$ is one to one.
I do understand what it means for a function to be one to one so I assume its basically the same for a transformation. However, I do not understand how this simple substitution converts the transformation into one that is one to one.
Given a transformation, how is one to tell if it is one to one?
If someone could explain this I would really appreciate it.
Typically, a function $f$ is one-to-one if it is injective. This means that if $f(x) = f(y)$, then $x=y$. For example, the function $(y_1, y_1)\mapsto (y_1, y_1y_2)$ is injective because if $(x_1, x_1x_2) = y_1, y_1y_2)$, then $(x_1,x_2) = (y_1,y_2)$. We can do the same test for any function. Here, a transformation just means function.