Probability theory: Relationship between random variables and events

Question

Those are the most basic concepts of probability theory, but how they are related / linked is rarely discussed in textbooks (at least those that I read).

An event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. (from https://en.wikipedia.org/wiki/Event_(probability_theory))

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. (from http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm)

Here are my thoughts. Would you agree? Any suggestions / references are very welcomed.

Consider as an example rolling dice twice. Random variable X is the sum of two rollings. So, for example, X({3,2}) = 5. I can say that {3,2} is my event that occured. In other words, random variable is a function that gets as input the specific event that occured. So, the input of random variable is a special case of event with only one outcome. The value of random variable (i.e., the output of random variable) will be associated with this specific event. Obviously, the same value of random variable can be associated with many events (e.g., X({3,2}) = 5, X({4,1}) = 5).

We can also consider probability of the random variable vs. probability of the event. Let’s say my event is A = {6,6}. So, P(X>11) = P(A). So, I can say that: “probability of random variable X to be more than 11 is equal to probability of event A (getting twice 6)”. In this example, the A contained only one potential outcome. But if I put it: P(X>5) = P(B), then B will contain many potential outcomes.

Thanks!

Answer 1

An event is a set of outcomes. You can specify certain events using random variables, but they are still events.

For instance, $\{X > 11\}$ is an event equal to the set $\{(6,6)\}$ containing a single outcome. $\{X=3\}$ is an event equal to the set $\{(1,2), (2,1)\}$ containing two outcomes.

So $P(X>11)$ is the probability of the event $\{X > 11\}$, which just happens to be described using a random variable.

Answer 2

I think this is a very good start! I'd like to make some minor changes here and there.

• An event is a subset of the sample space. You don't need to have a probability measure to define an event. As you learn more math, you'll find a new way to define random variables and see that a probability (measure) is just a special case of what is called a measure, but that comes much later.
• When you define the random variable $X$ as the sum of the values on two successive throws, your sample space does not contain the element $\{3,2\}$ (which imo is also bad notation, you're referring to an ordered pair, not a set of singletons) because that is not the outcome of the experiment where you observe sums. Instead, your sample space has the event $X=5$. If you look at the sample space of possible values obtained in two consecutive throws, that contains this ordered pair.
• When you say $\mathbb{P}(X>11)$, you mean to look at the probability of occurrence of all possible events where the sum of the values from two consecutive throws is greater than $11$, and the corresponding subset of the sample space is only the event that $X=12$.

tl;dr: I get what you mean but you need to understand better what a sample space is. A good place to start would be Feller's text on this "An Introduction to Probability Theory and its Applications: Part 1".