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.

  • $\begingroup$ I think it would be helpful to write out, in words, what this means, rather than rely solely on simpler notation. Sometimes notation is confusing for new students. I'd recommend updating your answer with a plain English description of the what your notation means. $\endgroup$ – StatsStudent Jan 15 '19 at 14:44
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    $\begingroup$ @StatsStudent Sure, good point; added. $\endgroup$ – Dougal Jan 15 '19 at 14:46
  • $\begingroup$ 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. $\endgroup$ – s5s Jan 15 '19 at 14:48
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    $\begingroup$ @s5s It means the minimum. The sum part is just pointing out that you can use the same kind of notation for sums (or the maximum, or products, or ...). $\endgroup$ – Dougal Jan 15 '19 at 14:49
  • $\begingroup$ @Dougal OK, so it means, once the random variables ${X_i}$ are realised, take the minimum realisation of the set? $\endgroup$ – s5s Jan 15 '19 at 14:50

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