Chapter 13 Integration and Probability
Maybe you’ve taken a courses in probability. Maybe you saw a little probability theory in high school. Maybe you’ve never studied anything to do with probability.
The first thing students usually learn is discrete probability, where the random variables take values in a finite set, with given probabilities for each outcome. That’s because this can be studied with middle school mathematics. For example, rolling two 6-sided dice leads to 36 possible outcomes, each equally likely; this in turn leads to 11 possible outcomes for the sum of the two dice, with probabilities ranging from \(1/36\) for 2 and 12 to \(6/36\) for 7. All questions about rolls of finitely many dice can be answered with careful thinking and basic arithmetic.
Random variables whose values are spread over all real numbers, or a real interval, require calculus even to define (much less to study). These are called continuous random variables, and are the topic of this section.
Philosophically, a real-valued random variable \(X\) is a quantity that has a value equal to some real number, but will have a different value each time some kind of experiment is run. It is unpredictable, therefore we cannot answer the question "What is the value of \(X\text{?}\)" but only "What is the probability that the value of \(X\) lies in the set \(A\text{?}\)"
For example, suppose we throw a dart at a 12 foot wide wall, from a long enough distance and with poor enough aim that it is as likely to hit any region as any other (if we miss completely, we get another try). Say the random variable \(X\) is the distance (in feet) from the left edge of the wall. We can ask for the probability that \(X \leq 2\text{,}\) that is that the dart lands within two feet of the left edge.
