Suppose I have a sequence of $n$ iid random variables (RV) $ X_1,X_2,...,X_n$ I want to denote the probability of at least one of those RV being lower than a threshold $x$. Is this notation correct?


Or does this mean probability of all RV being lower than $x$?

  • 2
    $\begingroup$ I would write $P(\min\{X_1,X_2,...,X_n\}<x)$ $\endgroup$ – callculus Mar 24 at 13:16

$P(X_1, X_2, \ldots, X_n <x)$ means all RV being lesser than $x$.

The desired event for which you want to denote its probability sounds ad hoc, so I wouldn't expect it to be only one way of denoting it. One way (perhaps the most intuitive) is as callculus suggests: $P(\min\{X_i \mid 1 \leq i \leq n\} <x)$. Another possibility is $P(\{i \mid X_i <x \} \neq \varnothing)$, or $P(\#\{i \mid X_i <x \} >0)$.


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