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?

up vote 35 down vote accepted

If the cdf of $X_i$ is denoted by $F(x)$, then the cdf of the minimum is given by $1-[1-F(x)]^n$.

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 Var, the probability $P(Y\leq y) = P(\min(X_1\dots X_n)\leq y)$ implies that at least one $X_i$ be 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 $y$.

If the $X_i$'s are iid, then the probability that all $X_i$ is greater than $y = (1-F(y))^n$. Therefore, the original prob is $1-(1-F(y))^n$.

Say $X_i \sim \text{Uniform} (0,1)$, then intuitively the probability $\min(X_1\dots X_n)\leq 1$ should be equivalent 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.

  • This site supports Markdown syntax for editing, and also LATEX for mathematical expressions. Further information can be found here: stats.stackexchange.com/editing-help. – chl Apr 28 '11 at 9:47
  • Thanks for providing the reasoning. I had a problem with non-identically-distributed variables, but the minimum logic still applied well :) – Matchu Mar 10 '13 at 19:56

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.

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    My opinion is that this book is THE book about extrem value theory – robin girard Jul 23 '10 at 21:56

This answer has been moved to here per recommendation in comments.

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    I think Nassim Taleb would approve of the general point. I would suggest adding more description into how your R code works exactly. – Andy W Mar 23 '16 at 16:51
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    I am puzzled by this post because it seems to focus on "estimates" and simulation, whereas the question is a purely theoretical one about random variables. Sure, the ideas are connected, but I cannot see how this post can be construed as an answer to the question. Without reading over the code very carefully, I haven't even any idea of what is being demonstrated or what is being plotted! – whuber Mar 23 '16 at 17:07
  • @whuber - This question and the answer were used to close other questions. The other questions, as I read them, were asking things that could be about application, not just theory. If you like I can delete it. If you like we can try an experiment in utility, and see how many up-votes it gets because a person was searching for this answer using suboptimal search terms. either works. – EngrStudent Mar 23 '16 at 17:20
  • If you have posted this as an answer to another question, why not post it in that other thread? If you think another question was incorrectly closed as a duplicate of this one, then please vote to reopen it. I am reluctant to interpret upvotes as indications of applicability: literally, they signify only that "this answer is useful" (of which I have little doubt), but not that it is necessarily on topic! – whuber Mar 23 '16 at 17:58
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    I like this post now that it has been explained with an edit. It says useful things about the sampling distribution of the minimum. Unfortunately, I don't think it can stand as an answer to the current question. If you're aware of another question that it could answer, then please consider moving it there. If not, consider asking the question that you had in mind when you did formulate this answer, and then answer your own question! Those options will enable your post to remain available and for people to appreciate it better. – whuber Mar 23 '16 at 18:52

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.

  • @gung I understand why you would conclude that, but this answer doesn't apply to the IID setting of the question: it therefore comes across as a (correct and potentially interesting) comment to the question itself. – whuber Dec 8 '17 at 17:37
  • It's up to you, @whuber, if you what to convert this to a comment, it's your call. – gung Dec 8 '17 at 18:29

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