Expectation

Notation

There's several different kinds of notation for the expectation that one might encounter.

First of all there is the different notations for the "E" part. It might be $\mathrm {E}$, $\mathbb {E}$, $\mathbf {E}$, or something close to that. If the random variable is clear from context, it might be denoted $\mu$.

Second, there are various subscripts that can accompany the "E" part. The main ones are:

• Random variable as the subscript: e.g. $\mathbf {E} _{X}(X+Y)$. The idea here seems to be to specify that it is not the joint expectation, but the expectation over some specific random variable. [1]
• Distribution as the subscript: e.g. $\mathbf {E} _{z\sim {\mathcal {D}}}f(z)$
• Parameter as subscript: in classical statistical inference, we are working with many probability measures (one for each value of the parameter $\theta$). So the subscript is used to specify which probability measure is being used to compute the expectation. e.g. $\mathbf {E} _{\theta }(X)$.