What is the probability distribution of summation of two independent random variables conditioned upon one of the summands? What would be $P(\Theta + W \mid \Theta)$ where $\Theta \perp W$. Intuitively I think It should be $P(W)$ shifted by $E[\Theta]$. Though I am not at all sure
If we condition it upon a specific $\theta$ i.e $P(\Theta + W \mid \Theta = \theta)$ then that is right as well since it is now actually a shifted $W$ only ($P(\theta + W))$).
I do not know how to go about finding $P(\Theta + W \mid \Theta)$  mathematically. Any help is much appreciated

Since $\Theta \ \bot \ W$, the distribution function for the sum is given by:
$$\begin{align}
F_{\Theta+W|\Theta}(t|r)
&\equiv \mathbb{P}(\Theta + W \leqslant t | \Theta = r) \\[6pt]
&= \mathbb{P}(r + W \leqslant t | \Theta = r) \\[6pt]
&= \mathbb{P}(r + W \leqslant t) \\[6pt]
&= \mathbb{P}(W \leqslant t-r) \\[6pt]
&= F_W(t-r). \\[6pt]
\end{align}$$
As you can see, this is indeed just a shifted distribution, where the distribution is shifted by the conditioning value for $\Theta$.
