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I don't understand the difference in notation between E( expression ) and E[ expression ] for [expected value][1]. The Wikipedia article seems to indicate E( ) is used for an arbitrary function of X, but I'm not sure what the difference in meaning is.

Google pointed me to an example of a usage difference [here][2], but the article didn't help to resolve my misunderstanding much:

In the following the operations E[ ] and Var[ ] (square brackets as opposed to parentheses before !) mean expected value and variance with respect to the structure function (prior distribution) U(η, θ) over HxΘ.

Also, what does it mean when expected value is applied as an operator?.

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    References [1] and [2] are missing.2012-09-01

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I would never mind about it. I personally use $E[X]$ as $X$ is a random variable. For its samples $x$ then I use the density for example $f_X(x)$, here ().

Expected value can also be though as the most likely sample (on average, where the average is taken with respect to the probability density function) from a stochastic process. For example if you have a standard normal Gaussian distributed $X\sim {\cal{N}}(\mu=0,\sigma=1)$ then you might expect that a given $x$ should be (most of the time) around $0$.

As an operator $E[X]$ is linear; such as $E[a_1X+b_1Y]=a_1E[X]+b_1E[Y]$. The name operator indicates that you apply it to some random variable or a group of them in a linear way.

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    @Did thank you. I got the point.2012-09-01