Sample two numbers from 1 to 10; maximize the expected product Assume you sample two numbers, randomly drawn from 1 to 10; you could choose two strategies: 1) pick with replacement and 2) pick without replacement. Which strategy would you prefer to maximize the expected product?
I encountered this problem while preparing for an interview. My intuition is picking with replacement would be better. However, I found it too complex to use the brute-force method to prove it. Could anyone suggest a better approach?

I think Glen_b's answer is wonderful intuition that you need to only consider the the covariance, as the unconditional expectations are the same. Below I prove the unconditional expectations are the same (ie $EX = EY$)
$$
\begin{align*}
EX &= \dfrac{1}{n}\sum_{i=1}^n i = \dfrac{n(n+1)}{2n} = \dfrac{(n+1)}{2}\\
E[Y|X=j] &= \dfrac{1}{n-1}\sum_{i \in [1,n] \setminus j} i = \dfrac{1}{n-1}\left(\dfrac{n(n+1)}{2} - j\right)\\
EY &= E[EY|X]\\
&= \dfrac{1}{n}\sum_{j=1}^n \dfrac{1}{n-1}\left(\dfrac{n(n+1)}{2} - j\right)\\
&= \dfrac{n(n+1)}{2(n-1)} - \dfrac{n+1}{2(n-1)} \\
&= \dfrac{(n+1)(n-1)}{2(n-1)} \\
&= \dfrac{(n+1)}{2}
\end{align*}
$$
