# minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$.

Is there any closed form expression or at least a good approximation for the distribution (I am most interested in the CCDF) of their minimum $Y = \min\limits_{1\leqslant i\leqslant N}(X_{i})$?

The CCDF of $Y$ is the product of the CCDF's of $X_1,\ldots, X_N$. The CCDF of $X_j$ (at nonnegative integer $x$) is $1 - \Gamma(1+x,\lambda_j)/x!$ where $\Gamma$ is the incomplete Gamma function. That's about as closed a form as you're going to get. As for approximations, which limit are you interested in?
• Thanks, Robert! I had the gamma formula as well. What I need to do is integrate this on $\lambda \in (0,\infty)$. Are there any good and integrable lower and/or upper bounds? Dec 13, 2012 at 21:14
• Integrate what exactly and over what domain? If you the $\Lambda$'s a reasonably large then Poisson can be approximated by Normal distribution, so if you were interested in, say, the expectation of the minimum it will be bounded between $\min_i(\Lambda_i)$ and $\min_i(\lambda_i-\sqrt{\log(N)}\sqrt{\Lambda_i})$. Dec 14, 2012 at 5:39
• Oops, I accidentally made some of the $\lambda$'s $\Lambda$. Dec 14, 2012 at 5:40
• Sorry, my integration explanation was very vague. I want to integrate the CCDF of the minimum Y, over variable $t\in(0,\infty)$, with $\lambda_j = \p_j t$. Unfortunately, the $p_j$'s are very small, so I'm guessing the normal approximation would not be so great here... Dec 14, 2012 at 11:56
• Do you really mean to integrate from $t=0$ to $\infty$? The CCDF goes to $1$ as $t \to \infty$, so the result would be $\infty$. Dec 14, 2012 at 18:43
For large $N$ asymptotics, you want to look into extreme value theory. In particular, take a look at this book.