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Given a set $R = (R_1, R_2, ..R_n)$ with $n$ positive random variables.

The sum is a fixed constant $C$.

What's the $E(min(R)+ max(R))$ ?

$R_1, ..R_{n-1}$ is uniform distributed from $(0, \frac{C}{n})$. $R_n$ is just an offset.

The $n-1$ variables are not necessary IID. There are some $k$ degrees of freedom among the $n-1$, and $k<n-1$. Assume the covariance matric is $\Sigma^{n-1 \times n-1}$, with $rank(\Sigma) =k$

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    $\begingroup$ The answer certainly depends on more than the information given when $n\ge3$. Compare for example when $R_1=\frac12$ and $R_2=R_3=\frac14$ almost surely, versus $R_1=R_2=R_3=\frac13$ almost surely. $\endgroup$ Apr 25, 2020 at 17:09
  • $\begingroup$ Imagine there is no constraint on the sum, then $E = \int p(x_1,..x_n) (\min(x1,..x_n)+\max(x1,..x_n)) dx_1...dx_n$. But how to include the constraint on the sum within this formalism? Is $E = \int p(x_1,..x_n) \delta(x_1+...+x_n-C)(\min(x1,..x_n)+\max(x1,..x_n)) dx_1...dx_n$ formally correct or not? $\endgroup$
    – Quillo
    Apr 25, 2020 at 17:38
  • $\begingroup$ sorry, I just updated with some extra information with $R$. $\endgroup$
    – peng yu
    Apr 25, 2020 at 18:06

1 Answer 1

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I suppose that you extract only $R_1...R_{n-1}$ and then $R_n$ is uniquely determined as a function of the $R_1...R_{n-1}$ values as $R_n = c-R_1-...-R_{n-1}$.

In this case you have to compute

$$ E_n = \int p(x_1,...,x_{n-1})[\min(x_1,...,x_{n-1}, c-x_{1}-...-x_{n-1})+ \\ +\max(x_1,...,x_{n-1}, c-x_{1}-...-x_{n-1})] dx_1...dx_{n-1} $$

where the integral is over the domain $[0,c/n]^{n-1}$.

For $c>0$, if the variables are i.i.d. over $[0,c/n]^{n-1}$, I obtain (using integration software) $$ E_2 = c/4, \quad E_3 = 7c/729, \quad E_4 = 11c/65536, \quad E_5 = 16c/9765625 $$ ...it seems it is going to zero quite quickly. Clearly a very partial answer but I hope it will give some insight.

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  • $\begingroup$ thanks! I think iid is not what i want here, there are some $k$ degrees of freedom among the $n-1$ , which $k<n-1$. maybe i should specify a Covariance matric of $\Sigma^{n-1 \times n-1}$ $\endgroup$
    – peng yu
    Apr 25, 2020 at 19:04
  • $\begingroup$ I don't think this is really useful to know. At the end you have to do this numerically for a generic $p(R_1..R_{n-1})$. The problem is formulated (in full generality) as in my answer, but, as you can see, then it is difficult to draw general conclusions if the joint probability $p(R_1..R_{n-1})$ is unknown. $\endgroup$
    – Quillo
    Apr 25, 2020 at 19:21
  • $\begingroup$ sorry for my misleading terms here, I believe i just specified the distribution, a multivariate uniform distribution. $\endgroup$
    – peng yu
    Apr 25, 2020 at 20:28
  • $\begingroup$ "multivariate uniform" ? I cannot find it, how is ti defined? $\endgroup$
    – Quillo
    Apr 26, 2020 at 11:32

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