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: $\text{E}[\mathbb{X}\space | \space\mathbb{X} \in \{1,2\}]$
Conditional Expectation Formula:$$\text{E}[\mathbb{X} | \mathbb{Y} = y] = \sum_x xf_{\mathbb{X}|\mathbb{Y}}(x,y)$$
so does:
$$\text{E}[\mathbb{X}\space | \space\mathbb{X} \in \{1,2\}] = 1*\frac{1}{3} + 2*\frac{1}{2} = \frac{4}{3}$$ since $\mathbb{X}$ is conditioned on a subset of itself?
Thanks!
Update: