Question

Are JOINT random variables 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

@Hero @jigglypuff314

Answer 3

No idea what it is you are trying to do.

If you wish to define marginal distributions, there might be one-tuples.

Why would x(1,H) = 6? What does that even mean?

Answer 4

A random variable is a funtion from the sample space to the real number.

I'm just merely defining X( (1,H)) to be 6

Answer 5

Okay, I don't know why you would do that. Just ANY Real Number?

Answer 6

yes any real number

Answer 7

Does this Random Variable have a Distribution?

Answer 8

it depends on the sample space

Answer 9

If you toss a fair coin, then the distribution is 1/2 for each outcome. But if it's not a fair coin, then the distribution is different.