Expected value of the sum of the two largest values from a Uniform parent

Question

Is the expected value of the sum of two greatest values in an uniform distribution in [0,1] of n random variables (x1,x2,x3,x4,...,xn) equal to E(max(x^n))+E(max(x^(n-1)))?

Answer 1

The probability that $n-1$ of them are less than $x$ is $nx^{n-1}(1-x)+x^n$, so the PDF of the second-highest is the derivative of that.
If the second-highest is $x$, then the expected value of the sum of the top one is $(1+x)/2$, and the sum of the top two is $(1+3x)/2$.
Integrate the PDF of the second-highest, multiplied by $(1+3x)/2$.

Answer 2

Let $X \sim \text{Uniform}(0,1)$ with pdf $f(x)$:

Let $X_n$ and $X_{n-1}$ denote the largest and second largest order statistics in a sample of size $n$. Then, the joint pdf of $(X_{n-1},X_n)$, say $g(x_{n-1},x_n)$, is:

where I am using the OrderStat function from the mathStatica package for Mathematica to automate, or do it manually following: Wiki -- Joint Order Stats

We desire $E_g[X_{n-1} + X_n]$ which is:

All done. As disclosure, I should perhaps add that I am one of the authors of the software used above.