Chapter 2 Section 1 Question 12

Question

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

Answer

Author: Mohammad-Ali Bandzar| Date:Oct 14 2020

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:$$\textit{I}_A(x)=\begin{cases}1&x\in\textbf{A}\\ 0&x\notin\textbf{A}\end{cases} $$Therefore no matter how we define our function $X$ we can always determine a set where for all $a\in A$ $\textbf{A}$ where $X(a)=1$ which fits our definition of the indicator function given above

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:$$\textit{I}_A(x)=\begin{cases}1&x\in\textbf{A}\\ 0&x\notin\textbf{A}\end{cases} $$Therefore no matter how we define our function $X$ we can always determine a set where for all $a\in A$ $\textbf{A}$ where $X(a)=1$ which fits our definition of the indicator function given above