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Why is $$P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$$

and similarly

$$P(\max\{X_1,...,X_n\}≤y)=P(X_1≤y, ..., X_n≤y)$$

I.e. why are $\min$ and $\max$ equivalent to AND (since $P(X_1≥y, X_2≥y)$ means $P(X_1≥y \text{ AND } X_2≥y)$ or $P(X_1≥y \cap X_2≥y)$) probabilities of all the r.v.s?

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    $\begingroup$ If the smallest value is bigger than $y$, then all the values are bigger than $y$. $\endgroup$
    – Glen_b
    Mar 12, 2016 at 8:59
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    $\begingroup$ To whom it may concern: This question is clearly on-topic here because it is about probability. I vote to keep it open. $\endgroup$
    – Sycorax
    Mar 12, 2016 at 17:57
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    $\begingroup$ Similarly if the largest value is less than y then all values are less than y. $\endgroup$ Nov 23, 2017 at 20:28
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    $\begingroup$ It perhaps worth adding that comma-separated events are considered to be connected with "and". $\endgroup$ Nov 23, 2017 at 21:47

2 Answers 2

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Because the events inside each $P(\cdot)$ are equivalent in both cases. To see this, let $X_{j} = \min\{X_{1}, \ldots , X_{n}\}$. Then if $X_{j} \geq y$, then it must also be that $X_{i} \geq X_{j} \geq y$ for all $i \in \{1,\ldots, n\}, i \neq j$. A similar argument can be used in the $\max$ case.

A standard mathematical statistics textbook will present this result when introducing order statistics.

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Consider four people, Yugi, Yuki, Yusei and Yuma. If we get their ages, observe that the following are equivalent.

  1. Yuma is younger than Yusei, Yuma is younger than Yuki, and Yuma is younger than Yugi (Analogous to: $y \le X_1, y \le X_2, y \le X_3$ where $y$ is the age of Yuma and the $X_i$'s are the ages of the others).

  2. Yuma is younger than the youngest among Yusei, Yuki and Yuma (Analogous to: $y \le \min\{X_1, X_2, X_3\}$).

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    $\begingroup$ Perhaps some additional elaboration could make it clear how this answers OP's question. $\endgroup$
    – Sycorax
    Mar 12, 2016 at 17:58
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    $\begingroup$ If you define the r.v.'s under consideration carefully, this isn't even an analogy -- they will be random variables. $\endgroup$
    – Sycorax
    Mar 12, 2016 at 18:03
  • $\begingroup$ @user777 what do you mean? For a given $\omega$, whatever is the value of $X_i$ in one side of the equation is the value of $X_i$ on the other side $\endgroup$
    – BCLC
    Mar 12, 2016 at 18:06

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