Re: How random is a random variable

On 2009-02-09 14:43:39 -0400, "Edward Jensen" <edward@xxxxxxxxxxxxxx> said:

"Gordon Sande" <g.sande@xxxxxxxxxxxxxxxx> wrote in message
news:2009020913265816807-gsande@xxxxxxxxxxxxxxxxx
Sorry for the confusion. Allow me to clarify. I am basicly asked how
equality between random variables are defined.
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?

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.

Your question seems to be about notation as once you have longer
descriptions
the difficulty disappears. Notation is about suitable short cuts and will
mean whatever you define it to mean. Sometimes notation can be helpful and
suggest additional things in a natural way and sometimes it gets in the
way.

Yes, I know that my confusion is probably due to notation, but what is the
conclusion then? Is X + X = 2X for any random variable X?

If your notation is that this means to add single numbers from a distribution
and there is only a single number for all the Xs on the page then X+X=2X

but

if your notaion is that each X is to be viewed as separate and connotes
a distribution then X + X will yield a convolution of the the X distribution
with itself.

Or maybe you have some other notational convention.

Which is correct depends on what YOU have DEFINED the notation to MEAN.

Often you will find that the notation is that X is the distribution but
x is a realization from it. But that is a choice of that authour. If the
choice is not spelled out then confusion might follow.

There are more than a few fields in mathematics where even "+" can have
a meaning that must be spelled out in great technical detail. Once you
get beyond school arithmetic notation can matter.


.

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