# what is the expected sum of max and min of n positive random variables?

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 $$k. Assume the covariance matric is $$\Sigma^{n-1 \times n-1}$$, with $$rank(\Sigma) =k$$

• 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. Apr 25, 2020 at 17:09
• 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? Apr 25, 2020 at 17:38
• sorry, I just updated with some extra information with $R$. Apr 25, 2020 at 18:06

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.

• 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 $\Sigma^{n-1 \times n-1}$ Apr 25, 2020 at 19:04
• 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. Apr 25, 2020 at 19:21
• sorry for my misleading terms here, I believe i just specified the distribution, a multivariate uniform distribution. Apr 25, 2020 at 20:28
• "multivariate uniform" ? I cannot find it, how is ti defined? Apr 26, 2020 at 11:32