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Say I want to distinguish the expected value for some random variable $\mathbb{E}(\eta)$ from its most likely / probable value.

Is there a conventional, succinct way to refer to the latter, other than: $MostProbableValue(\eta)=\cdots$ ?

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    $\begingroup$ Of course, such a value need not exist uniquely. For example, what would you use for a uniform distribution? $\endgroup$
    – MPW
    Mar 23, 2016 at 11:45
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    $\begingroup$ This is called the mode. $\endgroup$
    – TonyK
    Mar 23, 2016 at 11:46
  • $\begingroup$ um yes, I understand that (and it does exist uniquely for my random variable). I was just wondering whether there is an equivalent conventional notation to $\mathbb{E}$, but apparently not. $\endgroup$
    – maxheld
    Mar 23, 2016 at 11:48

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As already noted by MPW in the comments. The value is not unique. Also it does not need to exist (e.g. exponential distribution on $(0,\infty)$, which does not assume it's max on the definition range.). Hence a functional notation $MaxProb(P)$ would not be missleading.

On the other hand, a Maximum-Likelyhood-estimator of a statistic $\theta$ is often written as $\hat{\theta}$.

If you view the identity random variable $I: \Omega \rightarrow \Omega$, as statistic, you can write $\hat{I}$ for the most probable value.

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