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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. –  Did Sep 1 '12 at 12:01

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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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The wording "normal Gaussian" uses two words that are generally taken to have the same meaning: a normal random variable is a Gaussian random variable is a normal random variable. The follow-up ".. you might expect that a given $x$ should be around $0$" is false for normal random variables in general: only for normal random variables with means near $0$ and small variance should we expect the value to be near $0$. –  Dilip Sarwate Sep 1 '12 at 1:25
    
@DilipSarwate yes I meant "normal" as zero mean and variance $1$. I forgot the word standard. I edit now. Thanks for the comment. –  Seyhmus Güngören Sep 1 '12 at 11:12
    
Expected value can also be though as the most likely sample... Not. At. All. –  Did Sep 1 '12 at 12:00
    
@did Ok I think I should add on average. –  Seyhmus Güngören Sep 1 '12 at 12:03
    
To be sure you got the point, let me add an example: if X is Bernoulli, the event [X=E(X)] has probability zero, in fact, the event [E(X)-u<X<E(X)+u] also has probability zero, for some positive u, hence X is never around E(X). –  Did Sep 1 '12 at 12:07

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