Properties of the minimum of several random variables I've come across an interesting problem in my research that I don't quite know the answer to. Suppose I have a bunch of random variables:
$$ X_1, X_2, X_3, ... X_N $$
They are not identical but they are independently distributed. As a rule, they have different distributions. I am interested in the quantity:
$$ Z_N = \text{min}(X_1, ... X_N) $$
My first question: What can I say, in general, about the properties of $Z_N$? (Such as its expected value and variance) I understand that if the variables are IID, I can use Extreme Value Theory, but what if the variables have arbitrary distributions (say, different Gaussians?)
My second question: if all I have are samples of $X_i$, and not their distributions, and I get them one at a time online, is it possible to compute $E[Z_N]$ and $\text{Var}(Z_N)$ without storing all the samples?
 A: $$P\{Z_n > \alpha\} = P\{X_1 > \alpha, X_2 > \alpha, \cdots, X_n > \alpha\}
= \prod_{i=1}^n P\{X_i > \alpha\}$$
and so
$$F_{Z_n}(\alpha\} = 1-P\{Z_n > \alpha\} = 1 - \prod_{i=1}^n \left(1- F_{X_i}(\alpha)\right).$$
For the special case when the $X_i$ are exponential random variables
with parameter $\lambda_i$ (and so $P\{X_i > \alpha\} = e^{-\lambda_i \alpha}$),
we easily get that $Z_n$ is an exponential random variable with parameter
$\sum_{i=1}^n \lambda_i$.
More generally, taking the derivative of $F_{Z_n}(\alpha\}$ with respect to
$\alpha$ via the product rule, we get the density $f_{Z_n}(\alpha)$ as
$$f_{Z_n}(\alpha) = \frac{\mathrm d}{\mathrm d\alpha}F_{Z_n}(\alpha) = 
\sum_{i=1}^n \left(\prod_{j=1, j\neq i}^n \left(1- F_{X_j}(\alpha)\right)\right)f_{X_i}(\alpha)$$
which might at first glance seem to be a mixture of the $X_i$ 
densities, but really isn't
because the "weights" of the various densities are also functions of
$\alpha$ instead of being constants not depending on $\alpha$ and summing to $1$.
