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Here's the question:

Let $Z$ have CDF $F$ and pdf $f$ and let $A$ be a subset of the real line. Further, let

\begin{cases} W = 1 & \text{if $Z \in A$} \\ W = 0 &\text{otherwise} \\ \end{cases}.

a.) Find the PMF of $W$.

b.) Find the CDF of $W$.

For a.), I did this:

"Let $p = P(Z \in A)$. Then, the PMF for $W$ is $f_W(w) = p * \mathbb{1}(Z \in A) + (1-p)*\mathbb{1}(Z \notin A) = p = P(W = w)$."

Here is my work for b.):

"$F_W(w) = P(W \leqslant w) = \int_A f_W(w) = \int_A p * \mathbb{1}(Z \in A) = \int_A p dw$."

I'm really rusty with CDFs and PMFs, and have never taken a PMF/CDF of an identity operator of a random variable before. Where am I going wrong? Thank you!

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    $\begingroup$ $W$ is a binary variate, only taking two possible values.$0$ and $1$. It does not have a pdf (for a continuous measure) but a probability mass function with masses at $1$, $p$, and at $0$, $1-p$. Its cdf follows immediately. $\endgroup$ – Xi'an Jun 22 '20 at 13:41
  • $\begingroup$ Ah, it's a Bernoulli variable. I should have known. Thank you! $\endgroup$ – Bo Jack Jun 22 '20 at 13:52

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