About the correct definition of a binomial random variable

Question

I'm having trouble with how to exactly define a binomial random variable.

Let's fix a discrete probability space $(\Omega,P)$, where $|\Omega| \le \aleph_0$.

Then, if $X\colon \Omega \to \mathbb{R}$ is a real random variable, when exactly can we say that it is binomial ($X \sim Bi(n,p)$)?

I found two possible (non equivalent) definitions:

1. $X$ is the sum of $n$ independent Bernoulli random variables $X_1\colon \Omega \to \{0,1\},\dots,X_n\colon \Omega \to \{0,1\}$, where $X_1 \sim Be(p),\dots,X_n \sim Be(p)$;
2. The probability distribution of $X$ is given by $p_X(k)=P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$ when $k \in \{0,\dots,n\}$, while $p_X(x)=0$ if $x \in \mathbb{R} \setminus \{0,\dots,n\}$.

I know that 1) implies 2), but in general it is not true the converse, so that 1) is not equivalent to 2).

So maybe I should use 1) as a definition. Right?

I apologize for the low level of my question.

Thank you!

Answer 1

It's not a natural thing to do in probability to fix the sample space $\Omega$. The important thing is the random variable(s) and you can and should give yourself the freedom to modify the sample space as necessary to incorporate any new random variable(s) you want to discuss. This is discussed clearly and in detail in, for example, Terence Tao's 275A, Notes 0: Foundations of probability theory.

With the freedom to modify the sample space in mind, the two definitions are equivalent.

Said another way, the question is: what does it mean for two random variables to be "equivalent"? To my mind the cleanest answer is that they have the same distribution (not that they are literally the same function on the same sample space $\Omega$), or equivalently that they induce the same pushforward probability measures on $\mathbb{R}$. So when we define binomial random variables (or Gaussians or whatever else) we really only care about defining random variables up to this notion of equivalence, which does not depend on a choice of probability space. From this point of view the sense in which definitions 1) and 2) are equivalent is that they define the same probability measure on $\mathbb{R}$.

If you want to be really pedantic you can use definition 2) but I can't imagine a situation where the distinction would really matter anyway.