Bivariate random variables

[claudiocamera]

I am in a probability course, sometimes I come across with concepts that I know are easy, but unfortunatelly, I have dificult to handle it.

The question is :

If C and D are events and X and Y are Random Variables then:

C= min(X,Y) = P(X<x) U P(Y<y) ( WHY ????)

D=max(X,y) = P(X<x) AND P(Y<y) ( WHY ???? )

* AND is the interseccion operator.

 

[steve10]

cannot understand the equations. For example, the left hand side of

min(X,Y) = P(X<x) U P(Y<y)

is another random variable min(X,Y), while, in the right hand side, P(X<x) is a cummulative distribution function and so is P(Y<y). Both P(X<x) and P(Y<y) are function, but not random variables.

 

[claudiocamera]

Yes, it is really confused in the way it was presented , let's reformulate it.

Given C= min(X,Y) <2

It has as answer, the region in the XY plane corresponding to the events P(X<2) U P(Y<2) .

And given
D=max(X,Y) <2
It has as answer, the region in the XY plane corresponding to the events P(X<2) AND P(Y<2) .

* AND is the interseccion operator.

I just don't understand why. I think the problem is with the notation, what does the minimum of (X,Y) and Maximum of (X,Y) mean in the way presented ? I think that (X,Y) is the same that f(X,Y) but if it where, F(X,Y) would have only one result and not a maximum and a minimum. If it is not, what do they mean by (X,Y) ? I found the same notation in two books: Papoulis and Hsu, and honestly , I can't get the concept.