Probability & Statistics Ch.2 Section 1 Question 12

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Suppose X is a random variable that takes only the values 0 or 1. Must X be an indicator function? Explain.

Solution

Yes, X does have to be an indicator function.

The definition of an indicator function from Example 2.1.6: An indicator function is any function that can output only the values 0 and 1. If A is a subset of X for any set X, then the indicator function $\textit{I}_A: X \rightarrow \{0,1\}$ is defined by:

Therefore, no matter how we define our function X, we can always determine a set A where for all $a \in A$, we have $X(a)=1$, which fits our definition of the indicator function given above.