Let $X_i, Y$ be independent random variables both taking values in $\mathbb{N}$. Assume further that the $X_i´s$ are identical distributed. Then for $ 0 < A < \infty $ I want to compute the following sum.
$$ \mathbb{P}(\sum_{i=1}^Y X_i \leq A) $$
I would do this as follows:
$$ \mathbb{P}(\sum_{i=1}^Y X_i \leq A)= \mathbb{P}(\sum_{i=1}^Y X_i \leq A | Y = n) = \mathbb{P}(\sum_{i=1}^n X_i \leq A)= n \mathbb{P}(X_1 \leq A) $$
and thus
$$\mathbb{P}(\sum_{i=1}^Y X_i \leq A | Y) = \mathbb{P}(Y) \nobreakspace \mathbb{P}(X_1 \leq A) $$
which finally leads to
$$ \mathbb{P}(\sum_{i=1}^Y X_i \leq A) = \sum_{i=1}^{\infty} \mathbb{P}(Y=i) \nobreakspace i \cdot \mathbb{P}(X_1 \leq A) $$
Am I correct? I used Walds formula for inspiration.