Given a discrete random variable $\mathbb{X}$ on $\Omega = \{1,2,3\}$ with the following pmf:
$f_{\mathbb{X}}(1) = P(\mathbb{X} = 1) = \frac{1}{3}$
$f_{\mathbb{X}}(2) = P(\mathbb{X} = 2) = \frac{1}{2}$
$f_{\mathbb{X}}(3) = P(\mathbb{X} = 3) = \frac{1}{6}$
Find the following value of the conditional expectation: $\text{E}[\mathbb{X}\space | \space\mathbb{X} \in [1,2]]$
That is, I am wondering about the conditional expectation across a closed continuous interval.
Does the answer change from the previous thread? My initial thought would be no since $\mathbb{X}$ is discrete.
Conditional Expectation Formula:$$\text{E}[\mathbb{X} | \mathbb{Y} = y] = \sum_x xf_{\mathbb{X}|\mathbb{Y}}(x,y)$$