# What does $\min_{1 \le i \le n}X_i$ mean?

What does $$\min_{1 \le i \le n}X_i$$ mean?

In a context:

Show that the distribution function $$F$$ of $$\min_{1 \le i \le n}X_i$$ is ...

$$\min_{1 \le i \le n} X_i$$ means $$\min(X_1, X_2, \dots, X_n)$$, i.e. the minimum (smallest) of the $$X_i$$s.
The notation is analogous to $$\sum_{1 \le i \le n} X_i$$, which you'll sometimes see – as well as $$\sum_{i=1}^n X_i$$ – to represent $$X_1 + X_2 + \dots + X_n$$. You can use the same notation for the maximum ($$\max_{1 \le i \le n}$$), products ($$\prod_{1 \le i \le n}$$), unions of sets ($$\bigcup_{1 \le i \le n}$$), and many other operations.
• Sorry, it is unclear to me if it means the minimum $X_i$ or the sum of all $X_i$ from i=1 to i=n. I'm also unsure what "minumum $X_i$" would mean considering $X_i$ is a random variable. – s5s Jan 15 '19 at 14:48
• @Dougal OK, so it means, once the random variables ${X_i}$ are realised, take the minimum realisation of the set? – s5s Jan 15 '19 at 14:50