If $X_1, ..., X_n$ are independent identically-distributed random variables, what can be said about the distribution of $\min(X_1, ..., X_n)$ in general?
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asked Jul 20, 2010 at 5:23
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3 Answers
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If the cdf of $X_i$ is denoted by $F(x)$, then the cdf of the minimum is given by $1-[1-F(x)]^n$.
answered Jul 20, 2010 at 5:35
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If the CDF of $X_i$ is denoted by $F(x)$, then the CDF of the minimum is given by $1-[1-F(x)]^n$.
Reasoning: given $n$ random variables, the probability $P(Y\leq y) = P(\min(X_1\dots X_n)\leq y)$ implies that at least one $X_i$ is smaller than $y$.
The probability that at least one $X_i$ is smaller than $y$ is equivalent to one minus the probability that all $X_i$ are greater than $y$, i.e. $P(Y\leq y) = 1 - P(X_1 \gt y,\dots, X_n \gt y)$.
If the $X_i$'s are independent identically-distributed, then the probability that all $X_i$ are greater than $y$ is $[1-F(y)]^n$. Therefore, the original probability is $P(Y \leq y) = 1-[1-F(y)]^n$.
Example: say $X_i \sim \text{Uniform} (0,1)$, then intuitively the probability $\min(X_1\dots X_n)\leq 1$ should be equal to 1 (as the minimum value would always be less than 1 since $0\leq X_i\leq 1$ for all $i$). In this case $F(1)=1$ thus the probability is always 1.
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2$\begingroup$ This site supports Markdown syntax for editing, and also LATEX for mathematical expressions. Further information can be found here: stats.stackexchange.com/editing-help. $\endgroup$– chlApr 28, 2011 at 9:47
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$\begingroup$ Thanks for providing the reasoning. I had a problem with non-identically-distributed variables, but the minimum logic still applied well :) $\endgroup$– MatchuMar 10, 2013 at 19:56
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$\begingroup$ I think that answer 1-(1-F(x))^n is correct in special cases. Special cases is condition that pmf of r.v. is based on a formula for domain of r.v. If it be different in various parts of domain above mentioned formula deviates a little from actual simulation results. $\endgroup$ Nov 13, 2016 at 14:46
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$\begingroup$ Can anything be said about the mean and the standard deviation? $\endgroup$– vinntecApr 27, 2020 at 13:11
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Rob Hyndman gave the easy exact answer for a fixed n. If you're interested in asymptotic behavior for large n, this is handled in the field of extreme value theory. There is a small family of possible limiting distributions; see for example the first chapters of this book.
answered Jul 20, 2010 at 14:47
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2$\begingroup$ My opinion is that this book is THE book about extrem value theory $\endgroup$ Jul 23, 2010 at 21:56