Here we see another problem, the use of random. At the level of Kolmogorov, Loeve, Breiman, Neveu, Cinlar, Karr, etc., we use random only in random variable. We define random variable carefully but never say what random by itself means.
The MIT quantum mechanics lecture I mentioned keeps using random -- struggles with it, says it over and over as if that would make it more clear. In that lecture, the meaning of random is not clear.
What that prof means is likely
"the values of some random variables that are independent and identically distributed (i.i.d.)."
Then we can have some random variables that take values only on the positive, even integers. But, sure, they can't be uniformly distributed. So, with the values of such random variables, we can get some even integers selected independently.
In good treatments of probability, e.g., (from my bibliography file, with TeX markup)
Jacques Neveu,
{\it Mathematical Foundations of the Calculus of Probability,\/}
Holden-Day,
San Francisco.
independence is very nicely developed, e.g., to handle even a set of uncountably infinitely many independent random variables.
As I recall, Neveu was a Loeve student.
That book my Neveu is my candidate for the most elegantly written math book of all time.
But the development of independence is not trivial.
I would expect that independence is needed at times in physics, especially in quantum mechanics, but that only a small fraction of physics profs have studied independence much like what is in Neveu. One result is some confusion between independent and uncorrelated. When I'm watching a physics lecture and see such confusion, I stop taking the physics prof seriously and turn to something more serious, say, an old Marilyn Monroe movie!
The MIT quantum mechanics lecture I mentioned keeps using random -- struggles with it, says it over and over as if that would make it more clear. In that lecture, the meaning of random is not clear.
What that prof means is likely
"the values of some random variables that are independent and identically distributed (i.i.d.)."
Then we can have some random variables that take values only on the positive, even integers. But, sure, they can't be uniformly distributed. So, with the values of such random variables, we can get some even integers selected independently.
In good treatments of probability, e.g., (from my bibliography file, with TeX markup)
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco.
independence is very nicely developed, e.g., to handle even a set of uncountably infinitely many independent random variables.
As I recall, Neveu was a Loeve student.
That book my Neveu is my candidate for the most elegantly written math book of all time.
But the development of independence is not trivial.
I would expect that independence is needed at times in physics, especially in quantum mechanics, but that only a small fraction of physics profs have studied independence much like what is in Neveu. One result is some confusion between independent and uncorrelated. When I'm watching a physics lecture and see such confusion, I stop taking the physics prof seriously and turn to something more serious, say, an old Marilyn Monroe movie!