2
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

After thinking about the integral some more I believe that it may have to be split into three pieces.

$$ E[g(X,Y)]=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}g(x,y)f_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y \\= \int\limits_{-\infty}^{\infty}\int\limits_{y-t}^{y+t}xf_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y + \int\limits_{-\infty}^{\infty}\int\limits_{y+t}^{\infty}Cf_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y + \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{y-t}Cf_{X,Y}(x,y)\mathrm{d}x\mathrm{d}y $$

The first integral being justified by the fact that $g(x,y)$ takes on the value x over the area between the lines $y-t$ and $y+t$ and some random value C with PDF $f_C(x)=1/N$ everywhere else.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.