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Question,

Please provide any simplifications / bounds / concentration results for the expectation of the maximum of two correlated random variables.

$$E\left[\max\left(X,Y\right)\right]=\text{??}$$

$$E\left[\max\left(X,Y\right)\right]\leq\text{??}$$

$$E\left[\max\left(X,Y\right)\right]\geq\text{??}$$

Assumptions,

We can assume that $X$, $Y$ have a covariance $\sigma_{XY}$ and are from general distribution, $G\left(x\right)$ and $F\left(y\right)$ respectively.

We can assume that the mean, variance and higher moments exist for both the random variables.

Related Question on the Minimum,

Expectation of Minimum of Two Correlated Random Variables

If there are well known results, please provide references to papers that have detailed steps / derivations.

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