Word2Vec, softmax function
Your definition is correct. For the reference you can compare it with the probabilistic model from Tensorflow "Vector Representations of Words" tutorial:


  $$
\begin{align}
P(w_t | h) &amp;= \text{softmax} (\text{score} (w_t, h)) \\
\\
           &amp;= \frac{\exp \{ \text{score} (w_t, h) \} }
             {\sum_\text{Word w' in Vocab} \exp \{ \text{score} (w', h) \}}
\end{align}
$$


It's the same as yours, but they generalize the condition $h$ as a history. 

In case of skip-gram, the target word $w_t$ is any context word ($w_{-t}$ in your notation) and the history is the center word ($w_{t}$ in your notation). You specify $\theta$ explicitly in the condition, but that is usually omitted, because there is only one vocabulary in a given problem.