Question

Are JOINT random variable defined on a SAME sample space?

Answer 1

that is, GIVEN a sample space S. X and Y are called joint random variables means nothing more than X:S-->R and Y:S-->R?


Or is it the other way around? That is,

define X:S-->R, and Y:Ω-->R, where S is the sample space of X and Ω is the sample of Y.


Now, just in case my question isn't quite clear. Let me give an example.

Suppose I do an experiment. To toss a fair coin and roll a fair dice.

by definition of a sample space, that is all possible outcomes. So,
S = { (1,H), (2,H), (3,H), (4,H), (5,H), (6,H), (1,T), (2,T), (3,T), (4,T), (5,T), (6,T) }

and then I define random variables X and Y as follow:

X( (1,H) ) = 6
X( (1,H) = pi
..etc...

Y( (1,T) ) = 3
Y( (1,T) = sqrt(2)
...etc..

(I'm supposed to define X(s) for all s and Y(s) for all s, but I think you get the idea)


OR with the other definition
S = (H,T)
X(H) = 1
X(T) = 0

Ω = (1,2,3,4,5,6)
Y(1) = 2
Y(2) = 3
etc...


you see the difference? the random variables in the first examples takes a TWO-TUPLE ( e,g. (H,1)) as an input while the random variables in the second example take ONE-TUPLE as an input.

Answer 2

@mathmale @satellite73

Answer 3

In this case I have to admit I don't know the answer to that. In either case I'd suggest you look up "joint random variables" and then look for a reference there to "sample space." Good luck!

Answer 4

We're still struggling with the lack of an actual distribution.

Answer 5

ok, maybe here is an example. Say the experiment is you toss two fair dice. Then, the sample space is S= { (1,1), (1,2) ..., (6,6) } (all 36 of them)

let the random variable X be the sum
let the random variable Y be the maximum value of the two dice.

Answer 6

are X and Y joint random variables because they are defined on the same sample space S= { (1,1), (1,2) ..., (6,6) }