# Trying to rectify the differences between two different definitions of random variable

I am going through Introduction to Probability by Grinstead and Snell and apparently the book's definition of random variable is not conventional as discovered in another question I had here and I would like to understand their differences.

The book describes random variables and its distribution function as (page 27 of the PDF):

Let X be a random variable which denotes the value of the outcome of a certain experiment, and assume that this experiment has only finitely many possible outcomes. Let $$\Omega$$ be the sample space of the experiment (i.e., the set of all possible values of X, or equivalently, the set of all possible outcomes of the experiment.) A distribution function for X is a real-valued function m whose domain is $$\Omega$$ and which satisfies: $$m(\omega) \geq 0$$, for all $$\omega \in\Omega$$ and $$\sum_{\omega \in \Omega } m(\omega)=1$$

This seems to not line up with the Wikipedia definition here that describes random variables as a function $$X:\Omega -> E$$ where $$E$$ is usually $$R$$.

1. In the Wiki definition, it is correct to say that $$X$$ is not an outcome in $$\Omega$$ but instead maps $$\omega$$ to, usually, a real number? And is the probability mass function $$p_X(x)=P(X = x)$$ equivalent to my book's distribution function $$m(\omega)$$, except the domain of the probability mass function is all possible values of $$X$$, not $$\Omega$$?

2. Using the Wiki definition, is the event (say $$A$$) described by $$X = j$$ the set $$A= \{ \omega_1, \omega_2, .. \}$$ for all $$\omega \in\Omega$$ where $$X = j$$, such that $$P(X = j) = P(A)$$ (and $$P(X = j | E) = P(A |E)$$ if dealing with conditional probability conditioned on event $$E$$)?

3. In the continuous case, is there any practical difference between the two definitions? It seems if $$X$$ is viewed as a function, then $$X(x)=x\in\Omega$$ so would be equivalent as taking on an outcome from a continuous sample space?

1. Yes, it is correct to say that $$X$$ maps $$\omega$$ to, usually, a real number. Grinstead and Snell's $$m$$ is actually the probability measure of the probability space. (Minor quibble: Wikipedia misuses the verb "equip." It is $$\Omega$$ that is equipped with $$P$$, not the other way around.)
2. Yes, $$X=j$$ denotes the event $$\{\, \omega \in \Omega \mid X(\omega) = j \,\}$$.
3. In the continuous case, Grinstead and Snell abandon the idea that $$X$$ gives the outcome of the experiment. They regard $$\Omega$$ as a subset of $$\mathbb{R}^n$$, while $$X$$ takes values in $$\mathbb{R}$$. While I cannot speak for the authors, I suspect these choices were primarily driven by pedagogical expediency.
• Is the pmf of $X(\omega)$ functionally the same as $m$ in that both denote the probability of $X$, except that the domain of the two are different ($\mathbb{R}$ vs. $\Omega$)? Also, is the Wikipedia definition the preferred way of looking at random variables? I'm curious what the advantages of Grinstead and Snell's definition is, especially since starting from Chapter 4 they also start using the random variable as a function of ω. – Yandle May 15 '19 at 4:12
• Concerning the pmf vs. $m$ issue, note that the pmf of $X$ is defined on the range of $X$, which for a discrete random variable is not $\mathbb{R}$ by definition of discrete rv. – Fabio Somenzi May 15 '19 at 5:30
• Had another question that just came to me, for the continuous case (which I think also works for the discrete case?), per the Wiki definition, is the proper notation for the event where $X_1(\omega)<x_1,X_2(\omega) <x_2\dots,X_n(\omega)<x_n$ written as a set of n-tuple $\{\omega\in\Omega \space|\space (X_1(\omega)<x_1,X_2(\omega) <x_2\dots,X_n(\omega)<x_n)\}$? Each $X$ would still be a function of an outcome $\omega$ whose domain is now a subset of $R^n$? – Yandle May 21 '19 at 0:18
• Notation varies with authors. The one you used is rather common, but it helps understanding, in my opinion, to explicitly write $\{\omega \in \Omega \mid X_1(\omega) < x_1 \wedge X_2(\omega) < x_2 \wedge \cdots \wedge X_n(\omega) < x_n \}$. Each random variable $X_i$ is a function of the outcome $\omega$, which ranges over $\Omega$, whatever that is. That is, $X_i : \Omega \to \mathbb{R}$. – Fabio Somenzi May 21 '19 at 3:44