Is the location parameter always equal to $\operatorname{E}(X)$? Suppose that $X$ is a random variable such that one of the parameters of its distribution is a location parameter. Let's call this location parameter $\mu$ for convenience. Then is it always the case that $E(X) = \mu$?
Examples:

*

*Most famously, in a Normal$(\mu, \sigma^2)$ random variable, $\mu$ is the location parameter and we have $E(X) = \mu$.


*Suppose $X$ is a Laplace random variable. Then $X \sim \operatorname{Laplace}(\mu, b)$ where $\mu$ is the location parameter and again we have $E(X) = \mu$.
These are the two examples that immediately came to mind. I'm sure more examples can be found.
 A: It's not always the case that the mean defines the location parameter. For example the Cauchy distribution does not even have a mean. However, the median gives a well defined location parameter.
In addition, for extreme value distributions such as the Gumbel, the parameter referred to as the location parameter is not always equal to the mean (depending on the value of the shape parameter).
A: The reason $\mu$ appears in these specifications is that it is convenient to use the mean as one of the parameters when that is possible. In those cases $\mu$ will be the expectation, by definition.
A: For all symmetric distributions, the expectation, which is on the symmetry axis, is a natural choice for a location and it would be foolish to choose another.
For asymmetric distributions, you may have alternative options to define the location, such as the the median, the midrange, the mode... or even some parameter with no name, and all these may differ from the expectation.
As said by rubikscube09, location is an informal concept.
