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 ?
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By definition, $$ \mathbb P(Y=1) = \mathbb P(X>c) = e^{-c}. $$ From here is it trivial to compute the expectation. – Math1000 Jan 15 '19 at 22:24
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... providing that $c \ge 0$. Otherwise $\mathbb P(Y=1) = \mathbb P(X>c)= \mathbb P(X\ge 0) =1$ – Henry Jan 15 '19 at 22:26