Re: How random is a random variable

On Mon, 09 Feb 2009 17:51:48 +0100, Edward Jensen wrote:

"Lionel B" <me@xxxxxxxxxxx> wrote in message
news:gmphp9$6v$3@xxxxxxxxxxxxxxxxxxxx
On Mon, 09 Feb 2009 13:22:05 +0100, Edward Jensen wrote:

Hi.

I had the following semi-philosophical discussion with a colleague
today regarding how to interpret random varibles in equations. Let X
represent the outcome of rolling a fair die. Are 2X and X + X then
equal?

The problem is wheter to interpret X as a representative of the
underlying distribution or as a realization of the underlying
distribution.

Would you care to clarify what "representative of the underlying
distribution" and "realization of the underlying distribution" actually
mean? Not trying to be willfully obtuse here - I'm really not sure what
you intend by those phrases.

Sorry for the confusion. Allow me to clarify. I am basicly asked how
equality between random variables are defined.

Again, I urge you to read:

http://en.wikipedia.org/wiki/Random_variable#Equivalence_of_random_variables

If X is a random uniform
variable, how to you define X + X? Is it a triangle distribution or is
it merely a uniform distribution with a different support?

Note: "random variable" does not mean the same thing as "distribution". So you
can not ask "Is X + X a triangle distribution?" - that does not make sense. You
can ask "Does X + X have a triangle distribution?" (answer: no, it has a
uniform distribution; but if X_1 and X_2 are two independent uniform r.v.s
then X_1 + X_2 does have a triangle distribution).

The problem is actually as basic as how to view a random variable: as a
specific realization or as a function of all possible realizations. If
you view X as a realization then X + X = 2X would most certainly be true
since for any realization of X, called x, then x + x = 2x.

My question - which you have not attempted to answer - was: what do you *mean*
by "realization"? It is not clear to me.

On the other hand, if you view X as representing all possible
realizations (at the same time so to speak) then X + X = 2X will not
hold since the outcomes of X + X will occur with different probabilities
than for 2X (for example in the case of a discrete uniform
distribution).

I think you need a better idea of what is meant by "random variable". Any
decent textbook on probability theory should explain this. Short answer:
a random variable is a function on a sample space. As such, "equality" is
perfectly clearcut: the functions take the same value for every outcome
in the sample space. But note that there are other notions of "equivalence"
of random variables (eg. equality in distribution, almost sure equality,
etc.).

--
Lionel B
.

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