Suppose that we have two independent exponential variables (e.g. $X$ and $Y$) modulate a dependent variable (e.g. $Z$). So if two independent variables constitutes a joint pdf like $f(x,y)$, the question is, is it true to define $Z = f(x,y)$?
As a second question; Suppose that we connect two independent variables to a dependent variable by a 2D Gaussian function ($Z = exp(-(X^2/2)-(Y^2/2))$ ). In this case $X$ and $Y$ are independent variables because of $f(x,y) = f(x)f(y)$, given that $f(\cdot)$ is an exponential function. Now if we put $Z$ equal to a constant value, we provided the equation where $X$ and $Y$ are dependent together to satisfy equation. Is it a true argument for dependency of $X$ and $Y$ variables? Is it true that we've found a way to violate independencey of $X$ to $Y$ (or vice versa).
Thanks and apologize for asking naive question.
You can always define a function of two variables such as
$$z=f(x,y).$$
If $x$ and $y$ are two realizations of the random variables $X$ and $Y$, then you can define a new random variable $Z$, which we can symbolically denote as
$$Z=f(X,Y).$$
If $X,Y$ are independent wrt each other, they remain so (of course), and $Z$ is dependent on both (unless the function $f$ actually doesn't use its argument(s)).
You cannot "put $Z$ equal to a constant value", as $Z$ is a random variable. You can't "change" a random variable. You might compute the probability that $Z$ takes a given value, say $z$, and this equals the sum of the products of the probabilities of all $x,y$ such that $z=f(x,y)$.