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We study variants of the secretary problem, where $N$, the number ofcandidates, is a random variable, and the decision maker wants to maximize theprobability of success -- picking the largest number among the $N$ candidates-- using only the relative ranks of the candidates revealed so far.  We consider three forms of prior information about $\mathbf p$, theprobability distribution of $N$. In the full information setting, we assume$\mathbf p$ to be fully known. In that case, we show that single-threshold typeof strategies can achieve $1/e$-approximation to the maximum probability ofsuccess among all possible strategies. In the upper bound setting, we assumethat $N\leq \overline{n}$ (or $\mathbb E[N]\leq \bar{\mu}$), where $\bar{n}$(or $\bar{\mu}$) is known. In that case, we show that randomization oversingle-threshold type of strategies can achieve the optimal worst caseprobability of success of $\frac{1}{\log(\bar{n})}$ (or$\frac{1}{\log(\bar{\mu})}$) asymptotically. Surprisingly, there is asingle-threshold strategy (depending on $\overline{n}$) that can succeed withprobability $2/e^2$ for all but an exponentially small fraction ofdistributions supported on $[\bar{n}]$. In the sampling setting, we assume thatwe have access to $m$ samples $N^{(1)},\ldots,N^{(m)}\sim_{iid} \mathbf p$. Inthat case, we show that if $N\leq T$ with probability at least $1-O(\epsilon)$for some $T\in \mathbb N$, $m\gtrsim\frac{1}{\epsilon^2}\max(\log(\frac{1}{\epsilon}),\epsilon\log(\frac{\log(T)}{\epsilon}))$ is enough to learn a strategy that is at least$\epsilon$-suboptimal, and we provide a lower bound of$\Omega(\frac{1}{\epsilon^2})$, showing that the sampling algorithm is optimalwhen $\epsilon=O(\frac{1}{\log\log(T)})$.

Secretary Problems with Random Number of Candidates: How Prior Distributional Information Helps