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My question is how to calculate the following formula without iteration:

$$ \max \{A,B,C,D\} \tag 1 $$

suppose $A,B,C,D$ are normal and independent: I know (1) can be rewritten as

$$\max(\max(\max(A,B),C),D)$$

where

$$\max(A, B)= F(A)\cdot f(B) + F(B) \cdot f(A)$$

where $F$ is the CDF and $f$ is the PDF

if $$\max (A, B) = M_ab$$ then $$\max (A, B , C) = F(M_ab) \cdot f(C) + F(C) \cdot f(M_ab)$$ and so on.

Is there any other way for calculating maximum of $n$ random variables at once, instead of repetitively using pair-wise max?

(i.e $\max \{A, B, C, D\} = E$, where $E$ can be any type of distribution)

nauok
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2 Answers2

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If you have $n$ identically distributed and independent random variables each with

  • cumulative distribution function $F(x)$
  • and probability density function $f(x)$

then, by considering the probability that all $n$ of them are less than or equal to $x$ and then taking the derivative with respect to $x$, their maximum will be distributed with

  • cumulative distribution function $(F(x))^n$
  • and probability density function $n f(x) (F(x))^{n-1}$
Henry
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  • what if they were not identical? they just multiplied to each other? – nauok May 03 '19 at 14:27
  • could you please show me some references or examples? – nauok May 03 '19 at 14:28
  • @nauok For the CDF of independent non-identically distributed r.v.s you just multiply the individual CDFs. The pdf looks rather like a generalisation of Yves Daoust's answer. This is not difficult (it might be a book exercise): see for example https://math.stackexchange.com/questions/89030/expectation-of-the-maximum-of-gaussian-random-variables – Henry May 03 '19 at 14:44
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$$\frac{\max(A, B)}{F(A)\cdot F(B)}= \frac{ f(A)}{F(A)} + \frac{ f(B)}{F(B)}$$

seems to generalize nicely.