Independence of answers to queries sent to a random oracle
Firstly, note that saying that the queries $u_1,\ldots,u_Q$ are distinct implies that they are not independent, for example, if $u_i=a$ then $u_j\neq a$ for all $j&gt;i$.

Secondly, your argument indeed follows from the definition of a Random Oracle found in Stinson's book. Let $\mathcal{O}:\mathcal{X}\rightarrow \mathcal{Y}$ be a random oracle. That means that for every query $u$ that has not been queried before, it holds that $Pr[\mathcal{O}(u)=y]=1/|\mathcal{Y}|$ for every $y\in \mathcal{Y}$. Now, since $u_1,\ldots,u_Q$ are all distinct it follows that 

For all $y\in\mathcal{Y}$ and $i\in[Q]:  Pr[\mathcal{O}(u_i)=y]=\frac{1}{|\mathcal{Y}|}$ 

$\Rightarrow$ For all $(y_1,\ldots,y_Q)\in\mathcal{Y^Q}$:  $Pr[\mathcal{O}(u_1)=y_1,\ldots, \mathcal{O}(u_Q)=y_Q]= \prod_{i=1}^Q Pr[\mathcal{O}(u_i)=y_i]=\frac{1}{|\mathcal{Y}|^Q} $