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I am trying to work out a research problem I recently faced. I have a group of Poisson random variables and I want to find the distribution of the first sample that is equal to a specific number. In math terms I have
Given Poisson random variables $S_{1},S_{2}...S_{b}..$ that are generated from rate parameters $\beta_{1},\beta_{2}...\beta_{b}..$
Let $A=min_{b} \big( {S_{b}=s} \big) $
What is the probability of $\beta_{A}$ i.e $P(\beta_{A}=z)$
Given $z$, let $\mathcal{B}_z$ be the set of indices $b$ for which $\beta_b = z$. Then $\beta_A = z$ precisely when the first Poisson variable having the value $s$ has an index belonging to the set $\mathcal{B}_z$, that is,
$$ P(\beta_A = z) = P(A \in \mathcal{B}_z) = \sum_{b \in \mathcal{B}_z} P(A = b),$$
where the second equality holds as the events $\{ A = b \}$ are mutually exclusive for $b \in \mathcal{B}_z$. Then, $A = b$ if and only if the first $b - 1$ Poisson variables do not equal $s$ but the $b$th one does, $P(A = b) = P((S_1 \neq s) \cap \cdots \cap (S_{b-1} \neq s) \cap (S_b = s)) $. If you are willing to assume that the Poisson-variates are independent this can be further factorized and the the desired probability gets the form
\begin{align} P(\beta_A = z) &= \sum_{b \in \mathcal{B}_z} \left( \prod_{i=1}^{b-1} P(S_i \neq s) \right) P(S_b = s) \\ &= \sum_{b \in \mathcal{B}_z} \left( \prod_{i=1}^{b-1} \left( \sum_{j \neq s}^\infty \frac{\beta_i^j}{j!} e^{-\beta_i} \right) \right) \frac{\beta_b^s}{s!} e^{-\beta_b}, \end{align}
which is actually defined for all $z \in \mathbb{R}$ as if $\mathcal{B}_z = \emptyset$ we end up with an empty sum which equals zero.
