How would I define a random variable that is:
- Some random value (uniform distribution) if the magnitude of the difference of the parameters is greater than some threshold $t$
- Otherwise the random variable takes on the value of the first parameter
I'm thinking of something like this:
$$ g(x,y) = \left\{ \begin{array}{l l} x & \quad |x-y| < t\\ \text{Random real between $0$ and $N$} & \quad |x-y|>t \end{array} \right. $$
But I have no idea how to actually use this function, for example I want to calculate the expectation of $g(X,Y)$ but I can't figure out a way to do so using my current definition. Really my question is if it is possible to evaluate this integral given my definition of $g(X,Y)$: $$ E[g(X,Y)] = \int_{-\infty}^\infty \int_{-\infty}^\infty g(x,y)f_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y $$
where $f_{X,Y}(x,y)$ is the joint PDF of two continuous random variables $X$ and $Y$.
Any suggestions, or recommendations would be greatly appreciated!