Assume $X_1,X_2,\ldots,X_n$ are identical independent random variables, all distributed with $\text{Poisson}(\lambda)$ (same $\lambda$). I am interested to find out the distribution of their minimum: $X_\min=\min_i\{X_i\}$. Especially I am interested in the mean and variance of that random variable.
[Math] Minimum of identical independent Poisson random variables
probability distributions
Best Answer
$$E(X_{\mathrm{min}})=\sum_{k\geqslant1}P(X_{\mathrm{min}}\geqslant k)=\sum_{k\geqslant1}P(X_1\geqslant k)^n=\mathrm e^{-n\lambda}\sum_{k\geqslant1}\left(\sum_{i=k}^\infty\frac{\lambda^i}{i!}\right)^n$$