Minimum and maximum of series of random variables

I was reading this section about the minimum and maximum of a series of random variables:

Suppose that $$X_1, \dots, X_n$$ are independent variables with cdf's $$F_1, \dots, F_n$$, respectively.

The cdf $$F_m(\cdot)$$ of $$\max{\{ X_1, \dots, X_n \}}$$ is then given by

\begin{align} F_M(x) &= P(\max{\{X_1, \dots, X_n \}} \le x) = P(X_1 \le x, \dots, X_n \le x) \\ &= P(X_1 \le x \cap \dots \cap X_n \le x) = P \left( \cap_{i = 1}^n X_i \le x \right) \\ &= \prod_{i = 1}^n P(X_i \le x) = \prod_{i = 1}^n F_i(x) \end{align}

The cdf $$F_m(\cdot)$$ of $$\min{\{ X_1, \dots, X_n \}}$$ is given by

$$F_m(x) = 1 - \prod_{i = 1}^n (1 - F_i(x))$$

If I'm interpreting this correctly, the first part says that the probability that the maximum of a set of random variables is less than or equal to some value $$x$$ is equal to the probability that each of the random variables in that set is less than or equal to the value $$x$$? I'm struggling to understand why this must be true. After all, the former only considers a single value (the maximum of the set), and disregards the rest, whereas the latter considers whether every random variable is less than or equal to $$x$$, and so can have multiple random variables that satisfy this condition, no?

And with regards to the $$\min$$, where did the double $$1 -$$ in $$F_m(x) = 1 - \prod_{i = 1}^n (1 - F_i(x))$$ come from? Intuitively, I can see why you would require one $$1 -$$ in there to get the $$\min$$ from the $$\max$$, but I don't see why it does this twice?

Thank you.

• Consider a set of numbers $\{a_1 ... a_n\}$ and their maximum, denoted as $a^*$. If $a^* \leq x$, then of course all the elements of a set are also $\leq x$ because no element can be greater than maximum $a^*$. On the other hand, if $a_1 \leq x, ..., a_n \leq x$, then maximum $a^*$ is also $\leq x$. This establishes $a_1 \leq x, ..., a_n \leq x \iff a^* \leq x$. Using this with probabilities establishes the result. Mar 2, 2020 at 23:30

They are just using two basic facts : $$max \{a_1,a_2,...,a_n\} \leq x$$ iff $$a_i \leq x$$ for each $$i$$ and $$min \{a_1,a_2,...,a_n\} > x$$ iff $$a_i > x$$ for each $$i$$. This gives $$P(min \{X_1,X_2,...,X_n\}\leq x) =1-P(min \{X_1,X_2,...,X_n\} > x)=1-\prod P(X_i >x)=1-\prod [1-P(X_i \leq x)]=1-\prod (1-F_i(x))$$.