I have a problem with the next fact:

Let $X_1,...,X_n$ be random variables.

Why $$P\left(\min(X_1,X_2,...,X_n)>r\right)=P(X_1>r,X_2>r,...,X_n>r)?$$

Could you clarify me please? I can´t see very clear it, is part of a proof.

  • 2
    $\begingroup$ If all the values are larger than $r$, then the smallest one is (and vice-versa -- if the smallest one is larger than $r$, so are all the others). $\endgroup$
    – Glen_b
    Jan 26, 2018 at 14:14

1 Answer 1


For any $X_1,\dots,X_n$, whether or not exponential or independently distributed:

If $\min(X_1,X_2,...,X_n)>r$, then $X_1>r,X_2>r,...,X_n>r$.

If $X_1>r,X_2>r,...,X_n>r$, then $\min(X_1,X_2,...,X_n)>r$.

Therefore, the event $\min(X_1,X_2,...,X_n)>r$ occurs if and only if (i.e., is equivalent to) the event $X_1>r,X_2>r,...,X_n>r$ occurs. Therefore the probability of these events must be equal.


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