# How is the minimum of a set of IID random variables distributed?

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?

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 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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• 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
• 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. – Sasan Parsa Nov 13 '16 at 14:46
• Can anything be said about the mean and the standard deviation? – vinntec Apr 27 '20 at 13:11

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.

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