We draw $k$ balls from urn where balls are numbered from $1$ to $n$ (with returning). Let $X$-the smallest drawn ball, $Y$- the biggest one. Show that $\mathbb{E}X+\mathbb{E}Y=n+1$.
I've tried to set the distribution of that random variables. But then it was very hard to calculate the expectation number. What's more I've noticed that $\mathbb{E}X=\mathbb{E}Y$. I failed...
Do you have any idea how to show that?
If $Z:=X+Y$ then $Z$ and $2n+2-Z$ have equal distribution.
Consequently $$\mathbb{E}Z=2n+2-\mathbb{E}Z$$ hence $$\mathbb EZ=n+1$$
edit to make things more clear:
Let it be so that for $k=1,\dots,n$ ball with original number $k$ is also equipped with a second number $n+1-k$.
If we are looking at these second numbers then the largest drawn is $n+1-X$ and the smallest drawn is $n+1-Y$. We are dealing with a similar experiment so $(n+1-X)+(n+1-Y)$ will have the same distribution as $X+Y$.