Cross Validated Asked by user3285148 on November 27, 2020

Consider the following expected value

$$

E(max{a_1+epsilon_1, a_2+epsilon_2,…,a_n+epsilon_n})

$$

where the expectation is taken with respect to the random variables $(epsilon_1,…,epsilon_n)$ and $a_1,…,a_n$ are real numbers.

Can we say whether $E(max{a_1+epsilon_1, a_2+epsilon_2,…,a_n+epsilon_n})$ is greater or smaller than $max{a_1,…,a_n}$?

Can we characterise some upper or lower bound of $E(max{a_1+epsilon_1, a_2+epsilon_2,…,a_n+epsilon_n})$ that does not depend on the distribution of $(epsilon_1,…,epsilon_n)$?

No, consider the easiest case $n=1$. $E[a+epsilon]=a+E[epsilon]$ and it can be either greater or smaller than $a$ based on the distribution of $epsilon$.

Answered by gunes on November 27, 2020

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