Indicator function of an exponentially distributed random variable.

Question

If $X\sim\mathrm{Exp}(1)$ how do I have to interprete: $$ Y = \textbf{1}_{[c,\infty)}(X) $$ where $\textbf{1}$ is the indicator function. I realize that $Y$ is $0$ if $X < c$ and $1$ if $X >= c$. But how do I calculate the probability distribution of Y or even the expectation value ?