What is the difference between an indicator function and indicator random variable? Are they the same? My probability class uses the term indicator random variable, but it seems like indicator function is also used.
 A: They're the same thing. The reason for this is that formally, random variables are just functions. They map every single outcome in the sample space $\Omega$ to a real number.
For example, If $\Omega = \{ (1,1), (1,2), \cdots, (6,5), (6,6) \}$ is the set of the 36 different outcomes when rolling two distinguishable dice, then we can define a random variable $X : \Omega \to \Bbb R$ which maps each outcome to the sum of the two numbers rolled: $X( (n_1,n_2) ) := n_1 + n_2$. E.g. $X( (3,4) ) = 7$. 
An indicator random variable is a special kind of random variable, where the range of the function that corresponds to the random variable is $\{ 0, 1 \}$ instead of $\Bbb R$. So an indicator random variable is the same thing as an indicator function, and vice versa. 
In this case, we could define $A \subseteq X$ such that $A = \{ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \}$ (i.e. all outcomes where the numbers rolled sum to $7$). Then the indicator random variable $1_A$ is just a function $1_A: \Omega \to \{ 0,1 \}$ such that $1_A(a) = 1$ if $a \in A$ and $1_A(a) = 0$ if $a \notin A$. You could also call $1_A$ an indicator function. It doesn't really matter. An interesting property of indicator random variables / indicator functions is that $P(A) = E[1_A]$. So $P(A) = \frac{|A|}{|\Omega|} = \frac{1}{6}$. 
A: A random variable is a function that maps outcomes of a sample space to real values.
An indicator function has a codomain of binary values, $\{0,1\}$.   It is a piecewise function typically used to indicate whether an argument is within a given interval.   That is to say that for some set $\mathcal A$ with subset $\mathcal I$, then the indicator function $\mathbf 1_{\mathcal I}$ is defined such that for all $x\in \mathcal A$ then : $$\mathbf 1_\mathcal I(x) =\begin{cases}1 &:& x\in \mathcal I\\ 0 &:& x\notin \mathcal I\end{cases}$$
(Sometimes also written as $\mathbf 1_{x\in\mathcal I}$ )
When the domain happens to be the sample space (and thus interval is an event) then such an indicator function is called an indicator random variable.   Usually written in the form of $\mathbf 1_E$, it is the indicator that the realised outcome is within event $E$ ; that is, it has the value of $1$ if the event occurs, otherwise $0$.
