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I'm doing a question about a sequence of i.i.d. random variables that all have CDF $F(x) = 2^\alpha(2 - x)^{-\alpha}$ for $x < 0$ and $F(x) = 1$ otherwise. Without asking the whole question, basically, it asks about the CDF of the minimum of these random variables.

I thought about identifying a common distribution. To me, it looks like the beta distribution but the PDF for such a random variable raises $x$ to a power, unlike $F$.

I've also thought about complementary events (here). But from that, I don't see a way to apply the limit definition of the exponential function.

How should I think about the minimum of a sequence of i.i.d. random variables?

johnsmith
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  • Because $F$ is supported on the negative numbers, so is the minimum, whence there's no possible way that could be described by a Beta distribution, whose support is the interval $[0,1].$ Did you correctly describe your $F$? In any case, the axioms of probabilities imply the CDF of the minimum of $n$ iid variables having distribution $F$ is $1 - (1-F(x))^n,$ as is demonstrated in many threads here on CV: see https://stats.stackexchange.com/questions/129145 for instance. – whuber Jul 22 '23 at 15:43

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