Definition

A random variable is a function \(X\) which goes from the sample space to the real numbers: \[X : \Omega \rightarrow \mathbb{R}\]

Sample space \(\Omega\)

---

Example: Suppose we flip a fair coin 3 times. The outcome of a flip is heads (H) or tails (T). The sample space is \[\Omega = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\] Since the coin is fair, every outcome is equally likely. There are 8 outcomes, so the probability measure is \(P(\omega) = 1/8\) for every outcome \(\omega \in \Omega.\)

Let \(X\) be the number of heads flipped. Then \(X\) is a function from \(\Omega\) to \(\mathbb{R}.\) For example, \(X(TTH) = 1\) since there is one heads in \(TTH.\) Similarly, \(X(HHH) = 3\) since there are 3 heads in \(HHH.\)

Notation

Given a random variable \(X\) and a subset \(A \subset \mathbb{R},\) we can define a subset of \(\Omega\) as \[\{\omega \in \Omega : X(\omega) \in A\}\] This set is written \(X^{-1}(A).\)

If \(A\) is one point, \(A = \{a\},\) then \(X(\omega) \in A\) means \(X(\omega) = a.\) In this case, we write the set \(\{\omega \in \Omega : X(\omega) \in \{a\}\}\) as \(X = a\).

If \(A\) is an interval, for example \(A = (a, b],\) then we write the set \(\{\omega \in \Omega : X(\omega) \in (a, b]\}\) as \(a < X \leq b.\)

---

Using the coin flip example, let \(A = \{0, 1\}.\) Then \[\{\omega \in \Omega : X(\omega) \in \{0, 1\}\} = \{TTT, HTT, THT, TTH\}\] This is the set of all values in \(\Omega\) that have \(0\) or \(1\) heads. We could write this set in multiple ways including

• \(X^{-1}(A)\)
• \(X \in A\)
• \(0 \leq X \leq 1\)
• \(X < 2\)

The last 2 ways show more common notation that we will use going forward.

The set \(X = 2\) is \(\{HHT, HTH, THH\}.\) This is the set of values in \(\Omega\) that have \(2\) heads.

In this course, we will only look at events \(X \in A\) that are measurable. Remember that the probability measure might not be able to measure every event. In this class, we will take for granted that we can always compute \(P(X \in A).\)

---

Continuing with the coin flip example, we can compute the probability that the coin lands on heads \(0\) or \(1\) time. \[P(0 \leq X \leq 1) = P(\{TTT, HTT, THT, TTH\}) = \frac{4}{8} = \frac{1}{2}\] Next, we can compute the probability that \(X = 2,\) or that there are \(2\) heads. The probability the coin lands on heads twice is \[P(X = 2) = P(\{HHT, HTH, THH\}) = \frac{3}{8}\]