Indicator Functions: X, Y & Z

[quema]

Hi,

If $$X$$ and $$Y$$ are two random variables. Let $$Z=max\left\{{X,Y}\right\}$$. Can I write:



$$E(X-Y)max\left\{{X,Y}\right\}=E(X-Y)X1_{\left\{{X\geq{}Y}\right\}}+E(X-Y)Y1_{\left\{{X\leq{}Y}\right\}}$$

Do I need to assume that both random variables are independent?

Thanks.

 

[micromass]

Hi quema!

I think you can always write that! I'm not sure what you're trying to do, though...

 

[bpet]

Can I write:
$$E(X-Y)max\left\{{X,Y}\right\}=E(X-Y)X1_{\left\{{X\geq{}Y}\right\}}+E(X-Y)Y1_{\left\{{X\leq{}Y}\right\}}$$

Change the <= to < so you're not double-counting the case where X=Y, and it'll hold true regardless of dependence or whether they are discrete or continuous.