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I'm having trouble seeing why for RVs $X_{1}, \ldots, X_{n}$ it is true that:
$$Pr(min(X_{1}, \ldots, X_{n}) > x ) = Pr(X_{1} > x, \ldots, X_{n} > x)$$
In other words: Why is the event that the minimum of a set of RVs is greater than x equivalent to the event that all RVs are greater than x?
Is this because the minimum itself is a random variable and if one of the $X_{i}$ $\leq x$ it could happen that $X_{i}$ is the minimum?
Yup, the minimum is a r.v. too. Now, break the equation down into two parts:
First, suppose that $\min(X_1, \dotsc, X_n) > x$. Since $X_i \geq \min(X_1, \dotsc, X_n)$ for each $i$, it follows that $X_i > x$ for each $i$.
Second, suppose that $X_i > x$ for each $i$. Then $\min(X_1, \dotsc, X_n) > x$
So it must be that the event "$\min(X_1, \dotsc, X_n) > x$" is equivalent to the event "$X_i > x$ for all $i$". And since they're the same event, they must have the same probability.
